Practice library
Problems
1In how many ways can three rooks be placed on a chessboard of dimensions \(6 \times 2006\) so that no two of them attack each other? (Two rooks attack each other when they stand in the same row or in the …Open2Let \(T_1\) and \(T_2\) be the centroids of triangles \(A_1B_1C_1\) and \(A_2B_2C_2\), respectively. Prove that \[ \overrightarrow{A_1A_2} + \overrightarrow{B_1B_2} + \overrightarrow{C_1C_2} = 3\,\overrightarrow{T_1T_2}. \] …3Tomorrow's weather forecast consists of the following three claims: it will be cloudy, or it will snow, or the wind will blow; if it is cloudy and snowing, then the wind will blow; if the wind does not …4Let \(A\), \(B\) and \(C\) be finite sets whose sizes satisfy \[ |A \triangle C| + |B \triangle C| = |A \triangle B|. \] Prove that \(C\) is then trapped between the intersection and the union of the other …5For a natural number \(n\), let \(f(n)\) be the number written with the same digits taken in the opposite order (that is, read from right to left) whenever \(n\) is not divisible by \(10\); if \(10 \mid n\), …6Consider the set of five Serbian words (written in the Latin alphabet) \[ X = \{\ \text{aca},\ \text{konac},\ \text{lopte},\ \text{loto},\ \text{prst}\ \}, \] and define two relations on \(X\): for words …7It is known that the number \[ 21982145917308330487013369 \] is equal to \(n^{13}\) for some natural number \(n\). Determine \(n\).8Let \(a\), \(b\), \(c\), \(d\) and \(e\) be integers such that \[ a \ne b, \qquad b \ne c, \qquad c \ne d, \qquad d \ne e, \qquad e \ne a. \] Determine the smallest possible value of the expression \[ I = a^{2} + b^{2} + c^{2} + d^{2} + e^{2}. \] …9Determine the smallest natural number which, when divided by \(4\), \(6\), \(8\), \(10\) and \(12\), leaves the remainders \(2\), \(4\), \(6\), \(8\) and \(10\) respectively.10Let \(p\) be a number such that \(p\) and \(p^{2} + 2\) are both prime. Prove that \(p^{3} + 2\) is prime as well.11How many three-digit numbers written using the digits \(0, 1, 2, 3, 4, 5\) are divisible by \(15\), if (a) all digits must be distinct? (b) digits may repeat?12Let \(x\) and \(y\) be integers. Prove that if \(6x + 11y\) is divisible by \(31\), then \(x + 7y\) is divisible by \(31\) as well.13Determine all functions \(f : \mathbb{R} \to \mathbb{R}\) such that for every real number \(x\), \[ f(x+1) \le x \le f(x) + 1. \]14The villages \(A\) and \(B\) are \(3\) kilometres apart. There are \(100\) pupils living in village \(A\) and \(50\) pupils living in village \(B\). At what distance from village \(A\) should a school …15Let \(q\) be an odd integer. Prove that the equation \[ x^{3} + 3x + q = 0 \] has no solutions in integers.16Determine whether the number \[ 10^{5^{10^{5^{10}}}} + 5^{10^{5^{10^{5}}}} \] is divisible by \(11\).17Exactly \(2021\) points are chosen on the line \(AB\), and none of them lies on the segment \(AB\). Prove that the sum of the distances from these \(2021\) points to \(A\) can never be equal to the sum …18A triangle is cut into two triangles that are congruent to each other. Prove that the original triangle is isosceles.19A delegation of \(6\) people is to be chosen from a group of \(16\), consisting of \(4\) people from Serbia, \(4\) from Romania, \(4\) from Bulgaria and \(4\) from Macedonia. (a) In how many ways can this …20In a triangle \(ABC\), the altitude from \(A\) meets the line \(BC\) at \(D\), and its length satisfies \[ AD = \tfrac{1}{2}\,BC. \] Prove that the angle of the triangle at the vertex \(A\) cannot be obtuse. …21Determine the smallest six-digit number whose digits are all different and which is divisible by \(11\).22Solve the system of equations \[ \begin{aligned} x - y &= 2005, \\ \lfloor x \rfloor + \lfloor y \rfloor &= 2007, \end{aligned} \] where \(\lfloor t \rfloor\) denotes the integer part of the real number …23A circle is inscribed in triangle \(ABC\), whose sides have lengths \(BC = a\), \(CA = b\) and \(AB = c\). A line tangent to this circle meets the side \(AC\) at the point \(P\) and the side \(BC\) at …24Prove that no integers \(m\) and \(n\) satisfy \[ (m + n + 2)^2 = 3(mn + 1). \]25How many equivalence relations on a set of six elements have the property that every equivalence class contains at least two elements?26In a triangle \(ABC\), let \(C_1\) be the midpoint of the side \(AB\), so that \(CC_1\) is the median from \(C\). Let \(K\) be the midpoint of the segment \(CC_1\), and let the line \(AK\) meet the side …27Determine the greatest common divisor of the numbers \(2^{2006}-1\) and \(2^{2004}-1\).28Let \(X\) be a point in the interior of triangle \(ABC\), and let \(T\) be the centroid of that triangle. Points \(M\) and \(N\) lie on side \(BC\), points \(P\) and \(Q\) lie on side \(CA\), and points …29A natural number \(n\) leaves the remainder \(35\) on division by \(2009\), and also leaves the remainder \(35\) on division by \(2010\). What remainder does \(n\) leave on division by \(42\)?30Let \(O\) and \(H\) be the circumcenter and the orthocenter of a triangle \(ABC\), and let \(G_1\), \(G_2\), \(G_3\) be the centroids of the triangles \(HBC\), \(HCA\), \(HAB\), respectively. Prove that …31Let \(ABCD\) be a square and let \(E\) be the midpoint of its side \(CD\). The line through \(D\) perpendicular to the diagonal \(BD\) meets the line \(AE\) at a point \(F\). Prove that the points \(B\), …32Find the remainder on dividing the polynomial \(x^{2011} + 1\) by the polynomial \((x+1)^2\).33A row contains \(2016\) chairs. Each chair is to be painted either red or blue. In how many ways can this be done so that the number of neighbouring pairs of chairs painted in the same colour is even?34A park has the shape of a square with side \(1\) km. Inside it grow \(4567\) trees, each of diameter at most \(50\) cm, and each tree lies entirely within the park. Prove that the park contains a \(10\) …35In a quadrilateral \(ABCD\), \[ \angle ABC = 104^\circ, \qquad \angle ADC = 128^\circ, \qquad AB = BC = 2. \] Compute the length of the diagonal \(BD\).36Let \(AA_0\), \(BB_0\) and \(CC_0\) be the altitudes of a triangle \(ABC\), and let \(H\) be its orthocenter. Let \(M\) be the midpoint of the segment \(AA_0\) and let \(N\) be the midpoint of the segment …37Eight players took part in a chess tournament, and each of them played exactly one game against every other participant. A win earns \(1\) point, a loss \(0\) points, and a draw \(0.5\) points for each …38Let \[ N = 1^{n} + 2^{n} + 3^{n} + 4^{n}, \qquad n \in \mathbb{N}. \] What is the greatest number of zeros in which the number \(N\) can end?39Let \(a_{1}, a_{2}, \ldots, a_{n}\) be pairwise distinct numbers from the set \(\{1, 2, \ldots, n\}\), where \(n \in \mathbb{N}\). Prove that the number \[ (a_{1} - 1) + (a_{2} - 2)^{2} + \cdots + (a_{n} - n)^{n} \] …40Find all real numbers \(x\) for which \[ \bigl|\,|x| - 1\,\bigr| + \bigl|\,|x| + 2\,\bigr| = 3. \]41Find all pairs \((n, m)\) of integers for which \[ 3n^2 + 2nm + 3 = m^2 + 10. \]42Several numbers, all different from \(0\), are written on a board. Each of them is equal to half the sum of the remaining ones. How many numbers are written on the board?43Real numbers \(a_1 < a_2 < \cdots < a_n\) are given. Find every real number \(x\) for which the expression \[ |x - a_1| + |x - a_2| + \cdots + |x - a_n| \] takes its smallest value.44Let \(n\) be a natural number. Prove that \(3n^2 + 3n + 7\) is not the cube of any natural number.45Find the greatest common divisor of the two numbers \[ \underbrace{11111111}_{8}, \qquad \underbrace{11\ldots11}_{100}, \] written with eight and with one hundred digits \(1\), respectively.46A mosquito sits on the lower left cell of a rectangular board of format \(2003 \times 2004\). It travels above the board in the following manner: taking off from the cell it occupies, it flies over \(99\) …47Determine in how many ways the number \(441000\) can be written as a product of two factors \(m\) and \(n\) with \[ m > 1, \qquad n > 1, \qquad \gcd(m,n) = 1 , \] where the order of the factors is irrelevant, …48If \(a^{2} + a + 1 = 0\), what is the value of \[ a^{1995} + \frac{1}{a^{1995}} \, ? \]49Does there exist a triangle of area \(1\) whose sides \(b\) and \(c\) satisfy \(c \le b\) and \(b = 1.4\)?50Find all triples of pairwise distinct nonzero decimal digits \(a\), \(b\), \(c\) for which the fractions \[ \frac{\overline{ab}}{\overline{bc}} \qquad \text{and} \qquad \frac{a}{c} \] have the same value. …51Let \(S = \{s, i, c, g\}\). a) How many relations on \(S\) are not symmetric? b) How many antisymmetric relations are there on \(S\)?52The sum of \(49\) natural numbers equals \(999\). Find the largest possible value of their greatest common divisor.53Determine all values of the real parameter \(a\) for which the equation \[ \Bigl|\bigl||x-1|-2\bigr|-3\Bigr| = a \] has the greatest possible number of solutions.54Real numbers \(a\), \(b\), \(c\) satisfy the three inequalities \[ |b - c| \ge |a|, \qquad |c - a| \ge |b|, \qquad |a - b| \ge |c|. \] Prove that one of the numbers \(a\), \(b\), \(c\) is equal to the …55Let \(f\) and \(g\) be linear functions with the following three properties: the graph of \(x \mapsto f(g(x))\) passes through the point \((2021, 2022)\), the graph of \(x \mapsto g(f(x))\) passes through …56A polynomial \(P\) with integer coefficients satisfies \[ P\bigl(P(2023) + 2023\bigr) = 1. \] Which values can the number \(P(2023)\) take?57In the course of a five-year programme of study a student passed \(31\) exams in total. In every year he passed more exams than in the year before, and in the fifth year he passed three times as many exams …58Find all pairs of integers \(p\) and \(q\) for which the inequalities \[ p^{2} + q^{2} < 18p - 20q - 166, \qquad 32p - q^{2} > p^{2} + 12q + 271 \] hold simultaneously.59An angle of \(7^\circ\) is given. Using only compass and straightedge, divide it into seven equal parts.60Determine the smallest natural number the product of whose digits equals \(75600\).61Write each of the numbers \(1, 2, 3, \dots, 9\) into exactly one of the nine shapes in the figure - odd numbers into the triangles, even numbers into the squares - so that all \(12\) of the inequality …62Into a box, \(k\) smaller boxes are placed. Then \(k\) still smaller boxes are placed into some of the smaller boxes, each, and this procedure is repeated several times. If, at the end, \(m\) of all these …63In a trapezoid \(ABCD\) with \(AB \parallel CD\), the two angles at the base \(AB\) add up to \(90^\circ\). Prove that the segment joining the midpoints of the two bases has length equal to half the difference …64In triangle \(ABC\) the angle at \(B\) equals \(60^\circ\). The bisector of \(\angle CAB\) meets the opposite side at \(D\), the bisector of \(\angle BCA\) meets the opposite side at \(E\), and \(S\) is …65The quadrilateral \(ABCD\) is inscribed in a circle, and its diagonal \(AC\) is a diameter of that circle. Prove that the projections of the sides \(AB\) and \(CD\) onto the diagonal \(BD\) are equal.66Let \(a\), \(b\), \(c\) be positive integers such that all three of the numbers \[ p = b^{c} + a, \qquad q = a^{b} + c, \qquad r = c^{a} + b \] are prime. Prove that two of the numbers \(p\), \(q\), \(r\) …67Find all prime numbers \(p\), \(q\), \(r\), not necessarily different from one another, and all positive integers \(n\), for which \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{n}. \]68A car leaves town \(A\) and drives along a straight road at constant speed. Every \(15\) minutes it makes a turn of \(90\) degrees, to the left or to the right. Prove that the car can be back in \(A\) …69Natural numbers \(a\), \(b\) and \(c\) satisfy \[ a + \cfrac{1}{b + \cfrac{1}{c}} = \frac{4016}{2007} . \] Prove that \[ \cfrac{1}{c + \cfrac{1}{b + \cfrac{1}{a}}} = \frac{2007}{4016} . \]70It is known that \(60\) cows would eat all the grass in a meadow in \(24\) days, and that \(30\) cows would eat all of it in \(60\) days. Every day the same amount of new grass grows on the meadow. (a) …71Does there exist a bijection \(f \colon \mathbb{R} \to \mathbb{R}\) such that \[ f(f(x)) - f(x) = 56x + 2008 \] holds for every real number \(x\)?72Does the number \[ 2010^{2010} + 10^{2011} \] have more digits in its decimal representation than the number \(2010^{2010}\)?73In how many ways can \(11\) birds be placed into \(3\) identical cages so that every cage contains at least three birds?74For a natural number \(k\), let \(S(k)\) denote the sum of its digits. Do there exist natural numbers \(n\) and \(m\) such that \[ S(n) \cdot S(n+1) \cdot \ldots \cdot S(n+m) = 2011^{2010}\,? \]75Determine all natural numbers \(n\) for which the number of positive divisors of \(n^{3}\) is exactly \(2011\) greater than the number of positive divisors of \(n\).76Ten teams took part in a volleyball tournament, and each team played exactly one match against each of the other nine. When the tournament ended, the first team had \(x_{1}\) wins and \(y_{1}\) losses, …77At a round table sit \(2014\) people. Each of them either always tells the truth or always lies. Every single person at the table said the following sentence: "Apart from me and my two immediate neighbours, …78Determine how many distinct solutions the equation \[ \Bigl| \bigl| \cdots \bigl| \bigl| |x| - 1 \bigr| - 2 \bigr| - \cdots - 2016 \bigr| - 2017 \Bigr| = 2017 \] has, where the constants \(1, 2, \dots, 2017\) …79Let \(K\) be the midpoint of the side \(CD\) of a rectangle \(ABCD\). The lines \(BK\) and \(AC\) are perpendicular to each other and meet at the point \(H\), and \(G\) denotes the foot of the perpendicular …80Find all pairs of real numbers \(a\) and \(b\) such that the equality \[ \lfloor ax + by \rfloor + \lfloor bx + ay \rfloor = (a+b)\lfloor x+y \rfloor \] holds for all real numbers \(x\) and \(y\). (For …81Prove that in the regular octagon \(A_1A_2A_3A_4A_5A_6A_7A_8\) the diagonals \(A_1A_6\), \(A_3A_7\) and \(A_5A_8\) pass through one point.82Three distinct points \(A\), \(B\), \(C\) lie on a line \(\ell\), and a point \(O\) lies off \(\ell\). The perpendicular bisectors of the segments \(OA\), \(OB\) and \(OC\) form a triangle \(EFG\). Prove …83A snake starts in the upper-left cell of a \(2 \times n\) board, where \(n\) is a natural number. From one cell it may move to another whenever the two cells share an edge, but it may never visit a cell …84Let \(A\) and \(B\) be non-empty sets, neither of which is a subset of the other. For a natural number \(n\) consider the equality \[ \underbrace{A \setminus \bigl(B \setminus (A \setminus (B \setminus \cdots))\bigr)}_{n \text{ sets}} \;=\; \underbrace{A \mathbin{\triangle} \bigl(B \mathbin{\triangle} (A \mathbin{\triangle} (B \mathbin{\triangle} \cdots))\bigr)}_{n \text{ sets}} \] …85Let \(xOy\) be an angle, and let \(A\), \(B\) and \(C\) be points on the arm \(Ox\) such that \(OA = 3\), \(OB = 4\) and \(OC = 6\). Let \(D\) be the foot of the perpendicular dropped from \(B\) to the …86Determine all pairs of prime numbers \(p\) and \(q\) for which \((p^3 + 1)^q\) is the square of a natural number.87Aca and Branko play the following game on a \(2023 \times 2024\) board. First Aca chooses a square of the board and places a queen on it. Then the players move the queen alternately, following the rules …88Two people are talking. Person \(A\) says: "If we win at football, we will also win at basketball." Person \(B\) says: "If we do not win at basketball, we will win at football." Person \(C\) remarks: "At …89Let \(A\) be a subset of the set \(\{1, 4, 7, \dots, 1996\}\) containing exactly \(335\) elements. Prove that \(A\) contains two distinct numbers whose sum equals \(2000\).90Let \(a\) and \(b\) be arbitrary natural numbers, let \(M\) be their least common multiple and \(D\) their greatest common divisor. Prove that \[ a^n + b^n \le M^n + D^n \] holds for every natural number …91Find a five-digit natural number whose half is the square of a natural number and whose third is the cube of a natural number.92Prove that a natural number of the form \(4n + 1\) can be represented as a sum of two squares if and only if the number \(8n + 2\) can be represented as a sum of two squares.93The decimal representation of a positive integer \(n\) uses only the digits \(1\), \(3\), \(7\) and \(9\), and each of these four digits appears at least once. Prove that the digits of \(n\) can be rearranged …94A class has \(30\) students, and every day exactly three of them are on duty in the school kitchen. Prove that the duty roster cannot be arranged so that every two students of the class are on duty together …95Let \(S = \{1, 2, \ldots, 20\}\). What is the largest possible number of elements of a subset \(A \subseteq S\) with the property that \(2x \notin A\) whenever \(x \in A\)?96Let \(a\), \(b\) and \(c\) be the side lengths of a triangle, and set \[ p = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}, \qquad q = \frac{a}{c} + \frac{c}{b} + \frac{b}{a}. \] Prove that \(|p - q| < 1\).97Let \(n\) be a natural number. Prove that \(n^2 + 3n + 5\) is never divisible by \(121\).98a) Suppose the ordered quadruple \((x, y, z, w)\) is a solution of the equation \[ x^2 + y^2 + z^2 + w^2 = xyzw . \] Prove that \((yzw - x,\, y,\, z,\, w)\) is a solution of the same equation. b) Prove …99Find all pairs of real numbers \((x, y)\) satisfying \[ \frac{|x+y|}{1+|x+y|} = \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}. \]100Determine all natural numbers \(n\) for which the number \[ n^2 + 7n + 2 \] is equal to a product of several (at least two) consecutive natural numbers.101In the right triangle \(ABC\) the right angle is at the vertex \(C\). Let \(S\) be the midpoint of the side \(AB\), and let \(V\) be the point where the altitude dropped from \(C\) meets \(AB\). Determine …102Find every natural number with the following property: when the sum of its digits is added to the number itself, the result is \(313\).103Find the smallest natural number \(n\) for which the sum \[ n + 2n + 3n + \cdots + 9n \] is a number whose decimal representation has all of its digits equal.104Find every real number \(x\) that satisfies the equation \[ \bigl|\,2006 - |206 - x|\,\bigr| = 26 . \]105Prove that the inequality \[ x^2 + y^2 + 1 \ge 2\bigl(xy - x + y\bigr) \] holds for every pair of real numbers \(x\) and \(y\). When does equality occur?106In a convex hexagon \(ABCDEF\) the following lines are parallel: \[ AB \parallel FC \parallel DE, \qquad BC \parallel AD \parallel EF, \qquad CD \parallel BE. \] Prove that \(BE \parallel FA\).107Every cell of a \(3 \times 3\) board is to be painted in one of \(9\) colors so that all \(9\) colors are used. How many differently colored boards can be made? (Two boards are colored differently if they …108Two players alternately take balls from two boxes. On each turn, a player chooses one of the boxes and removes any number of balls from it (at least one). The player who takes the last ball wins. The first …109In how many ways can \(1000\) numbers be chosen from the set \(\{1, 2, \dots, 1999\}\) so that no two of the chosen numbers have sum \(1999\) or sum \(2000\)?110In how many ways can \(m\) distinct birds be placed into \(n\) distinct cages so that every cage contains at least one bird and at most two birds?111On the bisector of the angle \(\angle BAC\) of a triangle \(ABC\), points \(B_1\) and \(C_1\) are chosen so that \(BB_1 \perp AB\) and \(CC_1 \perp AC\). Let \(M\) be the midpoint of the segment \(B_1C_1\). …112Does there exist a natural number \(n\) for which the decimal expansion of \(n!\) has the form \[ n! = \ldots 2012\,\underbrace{00\ldots 0}_{k}, \] that is, ends in the digit block \(2012\) followed by …113Several lines are drawn in the plane. Line \(a\) intersects exactly three of the other lines, and line \(b\) intersects exactly four of the other lines. Line \(c\) intersects exactly \(n\) of the other …114Let \(d\) denote the greatest common divisor and \(v\) the least common multiple of the natural numbers \(m\) and \(n\). Prove that if \[ 3m + n = 3v + d , \] then \(n\) divides \(m\).115A rectangle \(ABCD\) has \(|AB| = 2a\) and \(|AD| = a\). Let \(E\) be the midpoint of the side \(AB\), and choose an arbitrary point \(F\) on the side \(AD\). The area of the triangle \(ECF\) depends on …116An equilateral triangle \(ACG\), a regular pentagon \(CDEFG\) and a regular octagon \(BHIJKLDC\) all meet at the point \(C\), as shown in the figure. Determine the angles of the triangle \(ABC\).117Two concentric circles of radii \(7\) cm and \(11\) cm are drawn in the plane. The smaller circle cuts a chord of the larger circle into three pieces of equal length. How long is that chord?118Find the smallest three-digit number with the property that every digit of its triple is even.119Let \(t\) be a real number and let \(a\) and \(b\) be positive real numbers satisfying \[ 2a^2 - 3abt + b^2 \;=\; 2a^2 + abt - b^2 \;=\; 0 . \] Determine the value of \(t\).120Find all values \(a \in \mathbb{R}\) for which the equation \[ |x - a| + |a - 1| = 1 \] has two solutions, and determine those solutions.121Words are built from the two letters \(A\) and \(B\) only. Is it possible to form a set of words containing \(3\) words of \(4\) letters, \(10\) words of \(5\) letters, \(30\) words of \(6\) letters and …122In a handball tournament every team played exactly one match against each of the other teams. A win is worth \(2\) points, a loss \(0\), and a drawn match gives \(1\) point to each of the two teams. The …123In an isosceles triangle, the bisector of one of the angles at the base is exactly twice as long as the altitude drawn to that base. Determine the angles of the triangle.124There are \(14\) books standing in a row on a shelf. In how many ways can \(5\) of them be chosen so that no two of the chosen books stand next to each other?125Let \(CD\) be the bisector of the angle \(BCA\) of a triangle \(ABC\), where \(D\) lies on the side \(AB\), and suppose that \[ AC + BD = BC + AD. \] Prove that the triangle \(ABC\) is isosceles.126A table of dimensions \(2010 \times 2011\) is given. Determine the largest number of cells that can be colored so that every \(2 \times 2\) square of the table contains at most two colored cells.127Let \(M\) and \(P\) be the feet of the perpendiculars from the vertex \(A\) of a triangle \(ABC\) to the bisectors of the exterior angles at the vertices \(B\) and \(C\), respectively. Prove that the length …128In a triangle \(ABC\) the angle at \(B\) is obtuse, \(\angle ABC > 90^\circ\), and the side \(AC\) is twice as long as \(AB\), that is \(2\cdot AB = AC\). Prove that \[ 2\cdot\angle ACB > \angle BAC. \] …129Let \(ABCD\) be a quadrilateral such that \[ \angle BCA + \angle CAD = 180^{\circ} \qquad\text{and}\qquad AB = AD + BC. \] Prove that \(\angle BAC + \angle ACD = \angle CDA\).130Consider strictly increasing sequences \(a_1, a_2, a_3, \dots\) of prime numbers in which any two consecutive terms differ by \(2\) or by \(4\); that is, \[ a_{i+1} - a_i \in \{2, 4\} \quad \text{for every } i. \] …131Let \(ABCDE\) be a convex pentagon whose five sides all have the same length. Suppose that two of its diagonals meet at an angle of \(60^\circ\). Prove that the pentagon has two parallel sides.132Let \(A\), \(B\), \(C\), \(D\) be four points in the plane, no three of them collinear. Every choice of three of these points forms a triangle, so the four points determine \(12\) angles in all. Write …133A mathematical commission has \(2n\) members, where \(n \geqslant 3\). Every member of the commission is in a quarrel with exactly one other member (the relation is symmetric). In how many ways can the …134Find all natural numbers \(n\) for which the three numbers \[ n-4, \qquad 2n+2, \qquad 4n+1 \] are all perfect cubes.135Find all three-element sets \(A\) with the following two properties: (i) the set \(A\) has at least two elements in common with its power set \(\mathcal{P}(A)\); (ii) \(3 \in A\). (It is understood that …136Let \(m > 1\) be a natural number. Prove that there is no sequence of \(2^{m}\) consecutive natural numbers all of which have exactly \(m\) prime factors, counted with multiplicity. (For example, the number …137Let \(n \ge 3\), and suppose \(n\) consecutive odd three-digit numbers are given. Prove that these \(n\) numbers can be arranged into a sequence \(b_1, b_2, \ldots, b_n\) so that the number \[ \overline{b_1b_2\ldots b_n}, \] …138The infinite sequence of natural numbers \(a_1, a_2, a_3, \dots\) is defined by \[ a_1 = a_2 = 1, \qquad a_{n+2} = a_{n+1} + a_n \ \text{ for every } n \in \mathbb{N}. \] Prove that \[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \] …139Two circles \(k_{1}\) and \(k_{2}\) intersect at two distinct points, and \(AB\) is their common chord. A point \(P\) is chosen on \(k_{1}\) so that it lies outside \(k_{2}\). The lines \(PA\) and \(PB\) …140The numbers \(1, 2, 3, 4, 5, 6, 7, 8\) are split into three disjoint nonempty sets. Let \(P_{1}\), \(P_{2}\) and \(P_{3}\) be the products of the numbers in the first, the second and the third set, respectively, …141Let \(n\) be the number \(100\ldots001\) whose decimal expansion consists of the digit \(1\), then \(2017\) digits \(0\), then the digit \(1\) again. Decide, with proof, whether \(n\) is divisible by (a) …142Show that the number \(7^{2018} + 9^{2020n}\) is divisible by \(5\) for every natural number \(n\).143Let \(x\) and \(y\) be real numbers satisfying \(y - x = 2\). Compute the value of the expression \[ 2x^3 + y^3 - 3y^2x + 6x^2 . \]144Find all pairs of real numbers \(x\) and \(y\) that solve the system \[ \frac{x}{6} + \frac{4}{y} = 2, \qquad \frac{18}{x} + \frac{y}{2} = 5 . \]145The caliph of Baghdad rewarded three wise men with ten purses: the first held \(0\) dinars, the second \(1\) dinar, the third \(2\) dinars, and so on up to the tenth, which held \(9\) dinars. The first …146The numbers \(1, 2, 3, 4, 5\) are divided into two groups so that each group contains at least one of them. Prove that one of the groups contains two numbers whose difference also belongs to that same …147Two operations \(F\) and \(G\) turn an ordered triple of real numbers into another triple by the following rules: \(F\) sends \((a, b, c)\) to \((a+1,\, b+c,\, c+1)\), and \(G\) sends \((a, b, c)\) to …148A convex quadrilateral \(ABCD\) satisfies \[ \angle DAB + \angle ABC = 120^\circ. \] Points \(P\) and \(Q\) are chosen so that the triangles \(ACP\) and \(BDQ\) are equilateral, with \(P\) lying in the …149For natural numbers \(m\) and \(n\), consider a board of dimensions \(m \times n\) made up of \(mn\) unit squares. Call the skeleton of the board the set of all unit segments that are edges of at least …150Determine every pair of integers \((x, y)\) for which \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} . \]151Let \(a\), \(b\), \(c\) be positive numbers with \(a^2 + c^2 = 2b^2\). Prove that \[ \frac{2}{a+c} = \frac{1}{a+b} + \frac{1}{b+c} . \]152Find all natural numbers \(m\) and \(n\) that satisfy \[ \frac{3}{m} + \frac{5}{n} = 1 . \]153The rhombus \(ABCD\) has an acute interior angle at the vertex \(A\). The perpendicular dropped from \(D\) to the side \(AB\) meets it at the point \(E\), which splits the side into the two pieces \[ |AE| = x , \qquad |EB| = y . \] …154Jana and Zana each wrote down her own age. Both ages turned out to be two-digit numbers written with the same two digits, only in the opposite order. Five years from now, Jana will be exactly twice as …155A competition paper had \(20\) problems. Each problem answered correctly was worth \(8\) points, each problem answered incorrectly cost \(5\) points, and a problem left blank scored \(0\). Tine handed …156Find every real number \(x\) that satisfies \[ \Bigl|\, \bigl|\, |2-x| - x \,\bigr| - 8 \,\Bigr| \le 2008 . \]157A competition paper consisted of \(24\) multiple-choice questions. A contestant who circled no answer to a question, or circled more than one, received \(0\) points for it; a circled correct answer was …158Find the smallest three-digit number with the following property: if its digits are written in the reverse order and the number so obtained is added to the original number, then every digit of the sum …159Find all pairs of integers \(x\) and \(y\) that satisfy the equation \[ x^2 + xy + y^2 = 1 . \]160Let \(E\) be a point of the diagonal \(AC\) of a rhombus \(ABCD\), with \(E \ne A\) and \(E \ne C\). Let \(N\) be the point of the line \(AB\) other than \(A\) for which \(EN = EA\), and let \(M\) be the …161Determine all triples of integers \((x, y, z)\) that satisfy \[ x^{2} + y^{2} + z^{2} = 2004\,xyz . \]162Can an equilateral triangle be divided - that is, actually cut up with scissors - into \(2006\) equilateral triangles?163One afternoon Ana and Olja each walked in a straight line to visit her boyfriend: Ana to Kosta's house, Olja to Laza's house. The two routes crossed at an old tree, and there the girls met. Standing under …164What is the largest number of chips that can be placed on the cells of a \(7 \times 7\) board so that no rectangle of area \(6\), with sides running along the grid lines, contains more than one chip?165A clock has three hands, each turning at its own constant speed: the second hand completes a full circle in one minute, the minute hand in one hour, and the hour hand in twelve hours. At midnight all three …166Let \(ABC\) be a triangle. The tangents to the circumcircle of \(ABC\) at the points \(B\) and \(C\) intersect at a point \(X\). The circle through \(A\), \(B\), \(X\) meets the line \(BC\) again at a …167Find all solutions of the equation \[ x = \bigl|\,2x - |60 - 2x|\,\bigr| . \]168In the right triangle \(ABC\) the right angle is at \(C\), and \(|AC| = 4\), \(|BC| = 8\). A point \(D\) is taken on the side \(BC\) so that \(|CD| = 5\). How far is \(D\) from the side \(AB\)?169The equilateral triangle \(ABC\) has sides of length \(4\) cm. Let \(D\) be the midpoint of the side \(AB\), and let \(E\) and \(F\) be the feet of the perpendiculars dropped from \(D\) to the sides \(BC\) …170Every school of a certain region sent exactly \(3\) students to a competition, and Andrej, Blaz and Zan all came from the same school. When the competitors lined up to collect their starting numbers, Andrej …171Ten teams took part in a volleyball tournament, and every team played exactly one match against each of the others. When the tournament ended, the first team had \(x_1\) wins and \(y_1\) losses, the second …172Two circles that do not intersect are given. Construct all of their common tangent lines.173Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]174Find every real value of the parameter \(a\) for which the polynomial \[ P(x) = x^{2021} - 2x^{2} + x + a^{3} - a \] is divisible by the polynomial \[ Q(x) = x^{2} - (a+1)x + a. \]175In a school there are \(2023\) pupils and \(2023\) lockers, the lockers bearing the numbers \(1, 2, \ldots, 2023\). At the start every locker is closed. The pupils file past the lockers one after another …176A binary relation \(\varrho\) on the set of real numbers is defined by \[ (\forall x, y \in \mathbb{R}) \quad x \varrho y \iff x^2 - 3xy + 2y^2 = 0 . \] (a) Determine whether \(\varrho\) is reflexive, …177On the set \[ A=\left\{0,\;1,\;-1,\;2,\;\tfrac12,\;-2,\;-\tfrac12,\;3,\;\tfrac13\right\} \] define the relation \[ \rho=\left\{(a,b)\in A\times A \;:\; \left(a^{2}-b^{2}\right)(ab-1)=0\right\}. \] (a) …178Prove or disprove the following assertion. Among any six positive integers it is always possible to choose three of them that are pairwise coprime, or three of them that have a common divisor greater than …179Let \(S\) be the midpoint of a segment \(AB\), and let \(C\) and \(D\) be points of the semicircle with diameter \(AB\) such that \(C\) lies on the arc \(AD\) and \(\angle CSD = 90^\circ\). Let \(E\) be …180Each unit cell of a \(3 \times 3\) table is coloured with one of three colours. How many such colourings are there in which every two cells sharing a side are coloured differently?181A bishop on a chessboard attacks every square lying on one of the two diagonals through it. Call a square covered if a bishop stands on it or a bishop attacks it. Prove that seven bishops can never be …182Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).183Let \(ABC\) be a triangle. On side \(AB\) choose points \(C_1\) and \(C_2\) with \[ AC_1 = \tfrac{2015}{3015}\,AB, \qquad AC_2 = \tfrac{2015}{3014}\,AB; \] on side \(BC\) choose points \(A_1\) and \(A_2\) …184Solve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.185Baron Munchausen lives in a country \(Z\) which has \(2018\) cities, some pairs of them joined by roads (every road can be travelled in both directions). The Baron has established that there is a city …186In the game Minesweeper, mines are placed on some cells of an \(a \times b\) board (\(a, b \in \mathbb{N}\)), and on every remaining cell one writes the number of neighbouring cells that contain a mine. …187Is it possible to divide a square into convex pentagons?188The cells of an \(n \times n\) table are to be coloured with \(n\) different colours in such a way that every row and every column contains cells of all \(n\) colours. Determine the smallest and the largest …189In a triangle \(ABC\) write \(a = BC\), \(b = CA\), \(c = AB\), and let \(S\) be the centre and \(r\) the radius of its inscribed circle. Consider the line joining the midpoints of the sides \(BC\) and …190Let \(X\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\) (\(AB \parallel CD\)). Prove that if \[ \angle ADX = \angle BCX, \] then the bisectors of the angles \(\angle ADX\), \(\angle DXC\) …191Find all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).192Jure wanted to prepare presents for nine friends, giving each friend two chocolate bars. In the shop he saw that a hazelnut chocolate bar costs \(6\) tolars more than a milk one, and that each of them …193Find all real numbers \(x\) that satisfy the inequality \[ \frac{|x-3| + x}{x+1} < 1 . \]194Find all pairs of natural numbers \(a\) and \(b\) that satisfy the equation \[ a^2 - 5ab + 24 = 0 . \]195The number whose cube equals \(2012^{12}\) was multiplied by the square of the number \(2012^{11}\). Which number was obtained? A \(2012^{58}\) B \(2012^{26}\) C \(2012^{88}\) D \(2012^{15}\) E \(2012^{12}\) …196A factory modernised its equipment, and its productivity then rose by \(25\%\). Some time later a number of workers were dismissed, and the productivity fell by \(20\%\). By what percentage has the productivity …197One digit of the seven-digit number \(2345678\) is to be deleted, so that the six-digit number left behind is divisible by \(9\). Which digit must it be? A \(8\) B \(7\) C \(6\) D \(5\) E \(4\)198Prove that a positive integer whose decimal representation uses no digits other than \(2\) and \(6\) cannot be written as a difference of the squares of two integers.199Find all natural numbers \(n\) for which the fraction \[ \frac{2n+3}{5n+7} \] can be reduced, that is, for which its numerator and denominator have a common divisor greater than \(1\).200Let \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]201Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).202The lengths of the sides of a triangle \(ABC\) are three consecutive natural numbers. The median drawn from \(A\) is perpendicular to the bisector of the angle \(\angle ABC\). Determine the lengths of …203Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).204The circles \(k_1\) and \(k_2\) meet at two points \(A\) and \(B\). Through \(A\) and through \(B\) two parallel lines are drawn. They meet the circle \(k_1\) for a second time at the points \(C\) and …205Let \(p\), \(q\), \(r\) be real numbers such that \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0 \qquad \text{and} \qquad p + q + r = 1 . \] Prove that for all real numbers \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …206Let \(x\), \(y\) and \(z\) be positive integers satisfying both \[ x^{3} - y^{3} - z^{3} = 3xyz \qquad \text{and} \qquad x^{2} = 2(y + z). \] Determine the value of \(x + y + z\).207Aca, Branka, Vera and Goran were each given the same kind of task by their mathematics teacher: divide one positive real number by another. Aca computed \(a_1 : a_2\), Branka computed \(b_1 : b_2\), Vera …208Suppose the numbers \(a\) and \(b\) satisfy \(a + b = 1\) and \(ab \neq 0\). Prove that \[ \frac{a}{b^3 - 1} - \frac{b}{a^3 - 1} = \frac{2(b-a)}{a^2b^2 + 3} . \]209Let \(M\) be the midpoint of the side \(BC\) and \(N\) the midpoint of the side \(CD\) of a rectangle \(ABCD\). Determine the ratio of the side lengths of \(ABCD\), given that \(AMN\) is a right triangle …210Nika and Tim played a series of card games. A draw was impossible. They agreed in advance that after each game the winner receives more points than the loser and that the loser receives a positive number …211Andrej and Blaz drew a circle in the yard and marked two diametrically opposite points on it. Each of them stood on one of these points, and at the same instant they started walking around the circle in …212Let \(a\) and \(b\) be arbitrary nonnegative real numbers. Prove that \[ (ab+1)(a+b) \ge 4ab , \] and determine all pairs \((a,b)\) for which equality holds.213The figure shows two squares and two congruent circles whose centres lie on a diagonal of the larger square. The smaller square occupies a corner of the larger one, and the diagonal in question runs from …214A rectangle \(ABCD\) satisfies \(|AB| = 10\) and \(|AC| = 5\sqrt{7}\). Let \(M\) be the midpoint of the side \(AB\). Inside the rectangle we draw the semicircle with diameter \(AB\) and the triangle \(CDM\), …215The squares \(ABCD\) and \(EFGH\) both have side \(1\). The first is cut into nine congruent small squares and the second into sixteen congruent small squares; in every small square the inscribed circle …216Lili noticed that the digits of the year \(2015\) have average \(2\), because \(\frac{2+0+1+5}{4} = 2\). How many years of the 21st century after \(2015\) have the same digit average as \(2015\)? A \(1\) …217Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]218Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).219A circle of radius \(\sqrt{2}\) is given. A second circle, of radius \(2\), has its centre on the first circle. Find the area of the shaded region, that is, of the part of the smaller disc that lies outside …220Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]221Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).222Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.223The numbers \(x\), \(y\) and \(a\) satisfy \(3^{x} = a\) and \(a^{y} = 81\). What is the value of the product \(x\cdot y\)? A \(4\) B \(3\) C \(12\) D \(0\) E \(1\)224Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …225Let \(a\) and \(b\) be natural numbers, both greater than \(1\), such that \[ \sqrt{a\sqrt{a\sqrt{a}}} = b . \] What is the smallest possible value of the sum \(a + b\)?226The real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).227In the right triangle \(ABC\) the right angle is at \(C\), the hypotenuse \(AB\) measures \(1\) dm, and \(\angle BAC = 30^\circ\). A point \(D\) inside the triangle satisfies \[ \angle BDC = 90^\circ \qquad\text{and}\qquad \angle ACD = \angle DBA . \] …228Nonzero real numbers \(a\), \(b\), \(c\) and a real number \(d\) satisfy \[ a = b + 2c, \qquad a + c = b + d, \qquad b = d + c . \] Which of the following equalities is certainly true? A \(d = 2c\) B \(a = 3c\) …229Peter keeps horses and cows on his farm. To begin with he had exactly as many horses as cows, and that common number was greater than \(0\). He then bought some more cows, so that the number of cows rose …230There are \(17\) girls and \(12\) boys on a playground. At least how many more children must arrive so that everybody present can then be divided into two groups of the same size, in such a way that each …231Every edge of a certain pyramid has length \(5\) cm, and the pyramid has five vertices in all, so its base is a square. All five vertices are sliced off, each by a single plane, and the cuts are made so …232How many three-element subsets \(\{a, b, c\}\) does the set \[ A = \{19, 20, 21, \ldots, 98\} \] have with the property that \(a + b + c\) is divisible by \(3\)?233A convex pentagon \(A_1A_2A_3A_4A_5\) is given. Let \(B_1\), \(B_2\), \(B_3\), \(B_4\) be the midpoints of the sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_5\), in that order, and let \(M\) be the …234A set \(A\) is given. Among its subsets a relation \(\sim\) is defined by \[ X \subseteq A, \quad Y \subseteq A, \qquad X \sim Y \iff X \cap Y \neq \varnothing . \] Determine whether \(\sim\) is reflexive, …235In a triangle \(ABC\) the angle at \(A\) measures \(60^\circ\). Write \(a\), \(b\), \(c\) for the lengths of the sides \(BC\), \(CA\), \(AB\). Prove that the area of the triangle equals \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …236Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]237Let \(x\), \(y\) and \(z\) be real numbers such that \[ x^{2} + y^{2} + z^{2} = 18 \qquad \text{and} \qquad xy + yz + zx = 9 . \] Determine the value of \(|x| + |y| + |z|\).238A circle is drawn through two vertices of a triangle and through the orthocentre of that triangle. Prove that this circle has the same radius as the circle circumscribed about the triangle.239(a) In how many ways can one choose two two-digit numbers that are not neighbours, that is, whose difference is not equal to \(1\)? (b) How many five-digit numbers are there in which the digit \(5\) occurs …240Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]241Let \(H\) be the orthocenter of a triangle \(ABC\), and let \(K\) be the point symmetric to \(H\) with respect to the midpoint of the side \(BC\). Prove that \(AK\) is a diameter of the circumcircle of …242At a volleyball tournament \(n > 1\) teams took part, and every two of them played exactly one match against each other. Prove that the teams can be numbered \(1, 2, \ldots, n\) in such a way that for …243Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.244A positive real number \(x\) is written on a board. In one move you are allowed to do the following: if a number \(a\) is already on the board, you may write down one of the numbers \(a+1\) or \(\frac{1}{a}\); …245Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.246Let \(A\), \(B\), \(C\) and \(D\) be finite sets such that \(D \subseteq A \cup B\), \(D \subseteq C\) and \[ |A \triangle B| + |B \setminus C| + |C \setminus D| + |B \cap D| = |A| . \] (a) Prove that …247Let \(a\), \(b\) and \(c\) be positive integers for which both of the numbers \[ 24^{a} + 2^{b} + 2018^{c} \qquad \text{and} \qquad 10^{c} + 3^{a} + 2018^{b} \] are divisible by \(7\). Prove that the number …248A pentagon \(ABCDE\) is inscribed in a circle. Let \(F\), \(G\), \(H\) and \(I\) be the midpoints of the segments \(BC\), \(CD\), \(DE\) and \(EA\) respectively. The lines \(FG\) and \(HI\) meet at the …249A relation \(\diamond\) is defined on the set \(\mathbb{R}\) of real numbers by \[ a \diamond b \quad \text{if and only if} \quad |a - 1| + |b - 2| \leqslant 1 . \] Suppose the real numbers \(x\) and \(y\) …250Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]251Let \(n\) be a positive integer. Let \(A_n\) be the set of all \(n\)-digit numbers whose decimal digits add up to \(4\), and let \(B_n\) be the set of all \(n\)-digit numbers whose decimal digits multiply …252Let \(\varrho\) be a binary relation on the set \(\mathbb{N}\) defined, for all \(x, y \in \mathbb{N}\), by \[ x \varrho y \iff (\exists k \in A)\ x + 2y = 3k \cdot x. \] In each of the cases \[ \text{(a) } A = \mathbb{N}, \qquad \text{(b) } A = \mathbb{T}, \qquad \text{(c) } A = \mathbb{Q}, \] …253Find all real numbers \(r\) for which there exists exactly one real number \(a\) such that the polynomial \[ p(x) = (x + a)\left(x^2 + rx + 1\right) \] has all of its coefficients nonnegative.254Let \(ABC\) be a triangle and let \(X\) be a point of its plane. Denote by \(A'\), \(B'\), \(C'\) the images of \(A\), \(B\), \(C\) under the reflection in the point \(X\). Let \(M\), \(N\), \(P\) be the …255Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]256Each of the numbers \(1, 2, \dots, 1995\) is to be given a sign \(+\) or \(-\). How should the signs be chosen so that the value of \[ \pm 1 \pm 2 \pm \cdots \pm 1995 \] is as close to zero as possible? …257It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.258Grandmother cut a round pizza into \(6\) equilateral triangles and \(6\) circular segments, as in the picture. Each of her \(6\) grandchildren ate one triangle. The grandchildren do not like the crust, …259For which values of the parameter \(a\) does the system of equations \[ |x - 1| + |y - a| = 1 , \qquad y = -2|x - 1| - 1 \] have exactly three solutions \((x, y)\)?260The edge of a wooden cube is a natural number \(a > 2\). The cube is painted all over and then cut into unit cubes. It turns out that the number of unit cubes with exactly two painted faces divides the …261Let \(ABCD\) be a quadrilateral and let \(K\) be a point inside triangle \(ABD\) such that triangles \(ABD\) and \(KCD\) are similar, the vertices corresponding in the written order. Prove that triangles …262In triangle \(ABC\) the sides satisfy \(|AB| = 2|AC|\). A point \(D\) is chosen on side \(AB\) and a point \(E\) on side \(BC\) so that \(\angle BAE = \angle ACD\). The segments \(AE\) and \(CD\) intersect …263Find all pairs of real numbers \(x\), \(y\) that satisfy the system \[ \begin{aligned} \frac{3}{x-4y} + \frac{2}{x+y-5} &= 0, \\ \frac{2}{x^{2}-4y^{2}} + \frac{1}{x^{2}+y^{2}-5} &= 0. \end{aligned} \]264During the first lesson of the day, the number of boys and the number of girls in a class were in the ratio \(3 : 4\). Before the second lesson another \(4\) girls joined the class and \(4\) boys left …265In how many ways can a king, a queen, two rooks, two bishops and two knights be placed on the eight squares of the first rank of a chessboard so that the two rooks stand on opposite sides of the king, …266Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.267Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]268Let \(M\) and \(N\) be two distinct points, neither of which lies on a given line \(p\). Construct a triangle \(ABC\) whose side \(AB\) lies on \(p\) and for which \(M\) and \(N\) are the feet of the altitudes …269Alenka and Barbara order a pizza. Two straight cuts, perpendicular to each other and neither of them passing through the centre of the pizza, divide it into four pieces. Alenka takes one piece first, then …270In the isosceles right triangle \(ABC\) the right angle is at \(C\) and each leg has length \(2\). A circular arc \(\ell\) centred at \(A\) divides the triangle into two parts of equal area, and a circular …271Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]272Let \(ABC\) be an isosceles triangle with apex \(C\). Points \(D\) and \(E\) lie on the sides \(AC\) and \(BC\) respectively, and the bisector of the angle \(\angle DEB\) and the bisector of the angle …273Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.274The natural numbers \(m\) and \(n\) satisfy \(19 \leq m \leq 49\) and \(51 \leq n \leq 101\). What is the largest value that \[ \frac{n + m}{n - m} \] can take? A \(20\) B \(30\) C \(40\) D \(50\) E \(60\) …275The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …276Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.277The lengths of the sides of a triangle are three consecutive natural numbers, each greater than \(3\). The altitude drawn to the middle side splits that side into two segments. Prove that the lengths of …278Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).279Find all composite natural numbers \(n\) which do not divide the product of all natural numbers smaller than \(n\), that is, all composite \(n\) for which \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …280The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).281Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.282Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.283Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).284Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.285An entry of a permutation is called right-minimal if it is smaller than every entry standing to its right. For example, in the permutation \[ (2,\; 1,\; 4,\; 6,\; 3,\; 7,\; 8,\; 5) \] the right-minimal …286For every point of the first quadrant, determine the line through that point which, together with the positive parts of the coordinate axes, bounds a triangle of the smallest possible area.287A cinema row has \(20\) seats. In how many ways can six couples take their seats in this row if every couple wants to sit on two adjacent seats?288Determine all positive integers \(n\) for which the number \[ \left| n - \sqrt{6 + \sqrt{6 + \sqrt{6}}} \right| + \left| 3 - \sqrt{6 + \sqrt{6 + \sqrt{6}}} \right| \] is rational.289Consider the polynomials \[ p(x) = x^3 + x^2 + x + 2 , \qquad q(x) = x^3 - x + 3 . \] Does there exist an integer \(m\) such that \(q(m)\) divides \(p(m)\)?290Two segments of lengths \(a\) and \(b\) are given. Construct a triangle \(ABC\) having these two segments as sides, in such a way that the angle opposite one of them is three times as large as the angle …291For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]292In a quadrilateral \(ABCD\) the sides \(AD\) and \(BC\) are equal, and the interior angles at \(A\) and \(B\) satisfy \[ \angle DAB + \angle ABC = 120^\circ . \] Prove that the midpoint of the diagonal …293The side lengths of a certain triangle are mutually distinct natural numbers, and its area is a natural number as well. Must that triangle be right-angled?294How many solutions does the equation \[ x - 2019\{x\} = 2019 \] have in the set of real numbers? Here, for a real number \(x\), the symbol \(\lfloor x \rfloor\) denotes the greatest integer not exceeding …295On the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that \[ AZ : ZB = BX : XC = 2021 : 2020. \] The perpendicular bisector of the segment \(XZ\) meets …296Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …297The number \(2025\) is written on a board. Ana and Bojan play the following game, moving alternately. A move consists of erasing the number currently on the board and writing in its place the difference …298The incircle of a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(D\) and \(E\) respectively. Let \(F\) be any point of the side \(AB\) lying between \(A\) and \(D\), and let \(G\) …299The real numbers \(a\) and \(b\) satisfy \[ \frac{3a}{a+b} + \frac{2b}{a+2b} = 1 . \] Determine every value that the expression \(\dfrac{2a-3b}{2a+b}\) can take.300Farmer Martin has made \(3\) identical bales of hay. The cross-section of each bale is a circle of radius \(r\). He stacks the bales into a pyramid, each bale touching the other two, and stretches a rope …301Find all three-digit numbers \(\overline{abc}\) that are divisible by \(9\) and satisfy \[ \overline{abc} = a^6 + b^2 + c^3, \] where \(a\), \(b\), \(c\) denote the digits of the number.302At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …303Vid cut a square \(ABCD\) of side length \(20\) units into \(400\) unit squares. Eva then picked four vertices of unit squares, all lying in the interior of \(ABCD\), that are the vertices of a rectangle …304At a national competition the students worked on \(4\) problems. Each problem was marked with a whole number of points, at least \(0\) and at most \(7\). Altogether \(42\) students competed. Exactly half …305Janez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …306A regular octagon is inscribed in a square of side length \(a\) so that four sides of the octagon lie on the four sides of the square. Express the side length of the octagon in terms of \(a\).307Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]308Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]309Find every natural number \(n\) with the following property: there is a rectangle whose side lengths are natural numbers, whose perimeter equals \(n\), and whose area is numerically equal to \(n\) as well. …310Consider the finite sequence of \(2003\) numbers given by \[ a_n = \left\lfloor \frac{n^2}{2004} \right\rfloor, \qquad n = 1, 2, \ldots, 2003, \] where \(\lfloor x \rfloor\) denotes the greatest integer …311Let \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …312Determine all natural numbers \(n\) for which the following assertion is true: a natural number \(x\) is divisible by \(n\) if and only if the sum of the digits of \(x\) is divisible by \(n\).313Ana, Biljana, Vesna and Gordana crossed a river in a canoe in the following way. There were three trips from the left bank to the right bank, and on each of them the canoe carried exactly two of the girls, …314In triangle \(ABC\) the bisector of the angle \(CAB\) meets the side \(BC\) at the point \(N\), and the bisector of the angle \(CBA\) meets the side \(AC\) at the point \(P\), where \[ PN = a . \] Let …315Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]316Can the plane be tiled by squares - that is, covered completely, with no two squares overlapping - in such a way that no side length is used by more than two of the squares?317Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …318Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]319Four vertices of a given regular octagon are to be coloured blue and the remaining four red. Two colourings are called equivalent if one of them is carried onto the other by a rotation of the octagon about …320Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).321The line through the circumcentre and the orthocentre of a triangle \(ABC\) (the Euler line of the triangle) crosses the interior of the side \(CA\) at a point \(M\) and the interior of the side \(CB\) …322Ana and Branko placed a number of tokens on the squares of an \(8 \times 8\) board, no square carrying more than one token. Ana then wrote down the number of tokens in each of the eight rows, and Branko …323Two roads run from Novi Sad to Belgrade, an old one and a new one, and they are joined by \(7\) connecting roads. In how many different ways can one travel from Novi Sad to Belgrade along these roads, …324Two circles touch each other internally at a point \(A\). Let \(AB\) be a diameter of the larger circle. Through the other endpoint \(B\) of this diameter a line is drawn which touches the smaller circle …325At most how many rooks can be placed on a chessboard of dimensions \(5 \times 4\) (five rows and four columns) so that every rook attacks at most one of the remaining ones? Here a rook attacks every rook …326Decide whether the following claim is true, and prove your answer. For every positive integer \(n\) there exists a positive integer \(x\) such that all three of the following hold: \(x\) is divisible by …327In the plane, two circles \(k_1\) and \(k_2\) and a line \(p\) are given. The line \(p\) cuts \(k_1\) at the points \(A\) and \(B\), and it cuts \(k_2\) at the points \(C\) and \(D\). Each of the two tangents …328Let \(n \geq 2\) be a natural number. Every cell of a square table \(A\) of size \(n \times n\) is filled with one of the numbers \(1\) and \(-1\). For each \(i \in \{1, 2, \ldots, n\}\) write \(k_i\) …329Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]330Let \(a\) and \(b\) be natural numbers. Prove that natural numbers \(c\) and \(d\) with \[ a^2 + b^2 + c^2 = d^2 \] exist if and only if at least one of the numbers \(a\) and \(b\) is even.331Two hundred real numbers are written around a circle. Their total sum equals \(200\), and the sum of any three numbers standing next to one another on the circle is at most \(3\). Is it possible for all …332An \(n \times n\) table is to be filled with zeros and ones so that for every index \(i \in \{1, 2, \dots, n\}\) the number of ones in the \(i\)-th row and the number of ones in the \(i\)-th column differ …333Let \(k\) be a circle with centre \(O\), and let \(T\) be a point outside it. The two tangents drawn from \(T\) touch \(k\) at the points \(A\) and \(B\). Let \(k'\) be the circle with centre \(T\) that …334Sixteen teams take part in a basketball tournament played as a double round robin: every two teams meet exactly twice. The eight best-placed teams qualify for the next tournament. Teams are ranked by the …335Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …336Every positive integer is painted in one of two colours, one of which is called red. The painting is periodic with period \(d\): the numbers \(x\) and \(x + d\) always receive the same colour. Suppose …337Let \(S\) be a finite set of natural numbers with the property that for every two elements \(x\) and \(y\) of \(S\) there exists an element \(z \in S\) such that \(z \mid x - y\). Prove that \(S\) contains …338Let \(x, y \in \mathbb{R}\) be such that \(x + y\) and \(x^2 + y\) are rational numbers. (a) If \(x + y^2\) is rational as well, must \(x\) and \(y\) be rational? (b) If \(x^3 + y\) is rational as well, …339Find all prime numbers \(p\), \(q\) and \(r\) for which the number \[ p^{\,q+r} + q^{\,p+r} + r^{\,q+p} \] is the square of an odd natural number.340For every natural number, Perica computed the remainder that this number leaves on division by the sum of its digits in the decimal system, and wrote that remainder on the board. Has Perica in this way …341Ana and Beno cycle clockwise along the rectangular track shown in the figure, on which the length of every section is marked. Ana rides the shorter of the two loops, Beno the longer one, and their speeds …342Jure wrote all the natural numbers from \(1\) to \(2015\) on a board. Urska then went through the numbers on the board from the smallest to the largest and rubbed out every one that was not divisible by …343In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …344Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.345An infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]346A quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).347In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).348For a natural number \(n\), determine the greatest common divisor of the two numbers \[ n^{2} + 1 \qquad \text{and} \qquad (n+1)^{2} + 1 , \] expressed in terms of \(n\).349Let \(ABC\) be an acute triangle. The circle \(k\) with diameter \(AB\) meets the side \(AC\) at \(M\) and the side \(BC\) at \(N\). The tangents to \(k\) at \(M\) and at \(N\) meet at the point \(P\). …350For every natural number \(n\), let \(x_n\) be the number obtained by writing the squares of the first \(n\) natural numbers one after another, in increasing order; for example \[ x_{12} = 149162536496481100121144 . \] …351Determine the smallest possible value of the expression \[ F = \max\{x,\, 1 - y\} + \max\{y,\, 2 - z\} + \max\{z,\, 3 - x\}, \] where \(x\), \(y\), \(z\) are real numbers, and find all triples \((x, y, z)\) …352Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …353Let \(ABC\) be a triangle and let \(a\), \(b\), \(c\) denote the lengths of the sides opposite the vertices \(A\), \(B\), \(C\) respectively. Prove that a point \(S\) is the centre of the inscribed circle …354How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?355Two equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) …356Prove that for every integer \(n \geqslant 2\) one can find \(n\) pairwise distinct positive integers whose squares add up to the square of a positive integer.357Let \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that \[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \] and …358Which of the following two numbers is greater: \[ \frac{1.\underbrace{11\ldots1}_{2005 \text{ digits}}}{1.\underbrace{11\ldots1}_{2006 \text{ digits}}} \qquad \text{or} \qquad \frac{1.\underbrace{0101\ldots01}_{4010 \text{ digits}}}{1.\underbrace{0101\ldots01}_{4012 \text{ digits}}} \, ? \] …359Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).360Let \(ABC\) be an isosceles triangle with \(AB = BC\). A point \(M\) is chosen inside it so that \[ \angle AMC = 2 \angle ABC , \] and a point \(N\) on the segment \(AM\) satisfies \(\angle BNM = \angle ABC\). …361Circles \(k_1\) and \(k_2\) intersect at points \(P\) and \(Q\), and \(k_1\) passes through the centre of \(k_2\). Distinct points \(A\) and \(B\) lie on the arc of \(k_1\) that runs inside \(k_2\), and …362The circle inscribed in triangle \(ABC\) touches the sides \(BC\), \(CA\) and \(AB\) at the points \(D\), \(E\) and \(F\) respectively. A point \(K\) lies on the same side of the line \(EF\) as the vertex …363Convex quadrilaterals \(ABCD\) and \(PQRS\) are given, where the vertices of the quadrilateral \(PQRS\) lie on the sides or in the interior of the quadrilateral \(ABCD\). Can the sum of the diagonals of …364Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]365The cells of a \(4 \times 4\) table are to be coloured with several colours so that in every figure congruent to the one shown below, all four cells have different colours. The figure may be rotated or …366Let \(ABC\) be a right triangle. Construct a point \(N\) inside \(\triangle ABC\) for which \[ \angle NBC = \angle NCA = \angle NAB. \]367Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.368On the sides of an acute triangle \(ABC\) points \(A_1 \in BC\), \(B_1 \in CA\) and \(C_1 \in AB\) are chosen so that \[ \angle CC_1B = \angle AA_1C = \angle BB_1A = \varphi, \] where \(\varphi\) is an …369Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]370Metka is standing \(60\) m east and \(80\) m south of the spot where Tine is standing. Both are the same distance from a linden tree in the town park, and the tree stands due east of Tine's spot. At the …371The value of the expression \(10^{2016} - 10^{15}\) is a natural number. What is the sum of the digits of that number? A \(1\) B \(17\) C \(2001\) D \(18\,000\) E \(18\,009\)372An airline charges a passenger nothing for baggage as long as its mass does not exceed a fixed permitted mass. Every kilogram above that must be paid for, always at the same price per kilogram. Mr and …373A real number \(a\) satisfies \(a^{2} - \tfrac{1}{2}a = \tfrac{1}{4}\). What is the value of \(a^{3} - \tfrac{1}{2}a\)? A \(-\tfrac{1}{4}\) B \(\tfrac{1}{4}\) C \(\tfrac{1}{2}\) D \(4\) E \(\tfrac{1}{8}\) …374Polona wants to draw three lines through one common point, as in the picture, so that the angles between them satisfy \(\beta = 2\alpha\) and \(\alpha = 3\gamma\). How many degrees must the angle \(\alpha\) …375The digits of a five-digit number add up to \(44\). What is the product of the digits of that number? A \(2^{3} \cdot 3^{8}\) B \(2^{3} \cdot 9^{3}\) C \(8 \cdot 4^{9}\) D \(8 \cdot 3^{4}\) E None of the …376Let \(x\), \(y\), \(z\) and \(w\) be natural numbers. At most how many of the six sums \[ x+y, \quad x+z, \quad x+w, \quad y+z, \quad y+w, \quad z+w \] can be odd? A \(2\) B \(3\) C \(4\) D \(5\) E \(6\) …377What is the value of \[ 2^{0^{2^{3}}} + 0^{2^{3^{2}}} + 2^{3^{2^{0}}} + 3^{2^{0^{2}}} \, ? \] A \(3\) B \(4\) C \(7\) D \(12\) E Greater than \(100\).378A goldsmith owns two alloys. The first is \(90\%\) gold, the second \(54\%\) gold. He melts together \(320\) g of the first alloy and \(160\) g of the second to obtain a new alloy. What percentage of gold …379For how many integers \(k\) is the number \(k + 6\) an integer multiple of the number \(k - 6\)? A \(0\) B \(4\) C \(6\) D \(8\) E \(12\)380Five semicircles, all of different sizes, stand side by side on one straight segment: each semicircle has its diameter on the segment, consecutive semicircles touch, and the five diameters together fill …381In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …382Let \(H\) be the orthocentre of an acute triangle \(ABC\), and let \(A_1\), \(B_1\) and \(C_1\) be the centres of the circles circumscribed about the triangles \(BHC\), \(CHA\) and \(AHB\) respectively. …383In the Mad Forest there lived \(6\) werewolves, \(17\) unicorns and \(55\) spiders. A werewolf can eat a spider or a unicorn, but not another werewolf; a spider can eat a unicorn, but neither a werewolf …384Let \(a\), \(b\), \(c\), \(d\) be positive real numbers such that \[ \frac{5a+b}{5c+d} = \frac{6a+b}{6c+d} \qquad \text{and} \qquad \frac{7a+b}{7c+d} = 8 . \] Determine the value of \(\dfrac{9a+b}{9c+d}\). …385Let \(a\), \(b\), \(c\) be positive numbers with \(a > c\) and \(b > c\). Prove that \[ \sqrt{c(a-c)} + \sqrt{c(b-c)} \;\leq\; \sqrt{ab} . \]386Prove that among all triangles with one and the same perimeter, the equilateral triangle has the largest area.387The three presents in the picture are all boxes in the shape of a cuboid with edge lengths \(10\) cm, \(20\) cm and \(30\) cm. Reading from left to right, the ribbon tying them measures \(x\) cm, \(y\) …388What is the largest natural number \(n\) with the property that, when \(n\) is divided by \(20\), the remainder is equal to the quotient? A \(21\) B \(92\) C \(231\) D \(399\) E \(440\)389Borut drew the table of size \(2 \times 7\) shown in the picture. He now wants to colour some of its cells so that every cell he leaves uncoloured shares a side with at least one coloured cell. What is …390Birds have gathered on a large pond. Exactly one third of them are swans and all the remaining birds are ducks. It turns out that \(70\%\) of all the birds on the pond are white, and every one of the swans …391Peter keeps \(111\) red and \(111\) blue marbles at home; they are made by his uncle. Every day Peter may visit his uncle and carry out one exchange: either he hands over \(11\) red marbles and receives …392Two teams, each consisting of \(6\) footballers, have at their disposal \(4\) pairs of shorts and \(4\) jerseys in each of the colours red, blue and white. In how many ways can the footballers dress for …393Find five real numbers whose pairwise sums are \[ 0,\; 2,\; 4,\; 5,\; 7,\; 9,\; 10,\; 12,\; 14,\; 17 . \] (Five numbers form exactly ten pairs, and ten values are listed; the sums may be listed in any …394(a) Factor the expression \(x^{4}+x^{2}y^{2}+y^{4}\) into a product of polynomials of lower degree. (b) Decide whether the number \(9^{1998}+3^{1998}+1\) is prime.395In a convex hexagon \(ABCDEF\), each of the two diagonals \(AD\) and \(BE\) divides the hexagon into two pieces of equal area. Prove that the quadrilateral \(BDEA\) is a trapezoid.396Let \(a\) and \(b\) be non-zero real numbers with \(a \ne -1\) and \(b \ne -1\), satisfying \[ \frac{a}{b+1} + \frac{b}{a+1} = 1 . \] Which of the following statements about the expression \(\dfrac{a}{b} + \dfrac{b}{a} - \dfrac{1}{ab}\) …397Five brothers - Jure, Klemen, Luka, Miha and Nace - bought a bar of chocolate. When they unwrapped it, they found that it had snapped into the seven pieces shown below, so they shared those seven pieces …398Let \(a\) and \(b\) be real numbers satisfying \[ \frac{a^2}{1+a^2} + \frac{b^2}{1+b^2} = 1 . \] Determine all possible values of the expression \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …399A merchant has to ferry seven goods across a river: a piece of cheese, a mouse, a rat, a cat, a dog, a wolf and a bear. His boat has room for only \(k\) of the seven at a time. If they are left without …400Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\), where \(n > 1\), be positive integers satisfying \[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}. \] Prove that \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) …401Let \(a_1, a_2, \ldots, a_n\) be positive real numbers with \(a_1 + a_2 + \cdots + a_n = 1\), and let \[ S = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i a_j}{a_i + a_j} \] be the sum of all \(n^2\) expressions …402Let \(N\) be the number whose decimal expansion consists of \(2025\) copies of the digit \(3\): \[ N = \underbrace{33\ldots3}_{2025} . \] What remainder does \(N\) leave when it is divided by \(12\)? A …403Squares \(BCDE\), \(ACFG\) and \(BAHK\) are erected outwards on the sides of a triangle \(ABC\). After that, the parallelograms \(BKPE\) and \(CDQF\) are drawn. Prove that the triangle \(PAQ\) is right …404In triangle \(ABC\) the angles at \(A\) and \(B\) measure \(\angle A = 50^\circ\) and \(\angle B = 60^\circ\). Points \(D\) and \(E\) are taken on the sides \(AB\) and \(BC\) respectively, so that \[ \angle DCA = \angle EAC = 30^\circ . \] …405Miljan and Mladen play the following game. They take turns naming divisors of \(200\), with one restriction: the number a player names must not be a divisor of any number named earlier in the game. A player …406The points \(A, B, C, D, E\) lie on one circle in such a way that \(A\) and \(D\) are on opposite sides of the line \(BC\), and \(B\) and \(E\) are on opposite sides of the line \(CD\). Given that \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …407Two points \(A_1\), \(B_1\) and a line \(p\) are given in the plane. Construct a triangle \(ABC\) in which \(A_1\) is the midpoint of the side \(BC\), \(B_1\) is the midpoint of the side \(CA\), and the …408A number is written in every cell of an \(8 \times 8\) table. A move consists of choosing any \(3 \times 3\) square of the table (nine cells) or any \(4 \times 4\) square (sixteen cells) and increasing …409A triangle \(ABC\) has \(AB = 2\), \(BC = 3\) and \(CA = 4\). Find a polygonal line \(XYZ\) whose endpoints \(X\) and \(Z\) lie on the boundary of the triangle \(ABC\), such that \[ XY = YZ = 1 \] and …410Let \(ABC\) be a triangle with \(BC \neq CA\), and let \(H\), \(T\) and \(O\) be its orthocentre, centroid and circumcentre. Let \(P\) be the point symmetric to \(T\) with respect to \(O\), and let \(Q\) …411A plane figure of area greater than \(1006\) can be placed inside a rectangle with side lengths \(2011\) and \(1\). Prove that the figure contains two points, on its boundary or in its interior, whose …412Let \(ABCD\) be a convex quadrilateral which is not a trapezoid. The perpendicular bisectors of the sides \(AD\) and \(BC\) meet at a point \(P\), and the perpendicular bisectors of the sides \(AB\) and …413Congruent circles \(k_1\), \(k_2\) and \(k\) all pass through a point \(P\), and each pair of them meets in one further point: \(k\) and \(k_1\) meet again at \(A\), \(k\) and \(k_2\) meet again at \(B\), …414In a triangle \(ABC\) the bisector of the angle at the vertex \(A\) meets the side \(BC\) at the point \(D\). The perpendicular dropped from \(B\) to the line \(AD\) meets the circumcircle of the triangle …415A cinema hall has \(2015\) seats, and \(2014\) viewers, one of whom is Mika, walk in. They all sit down on arbitrary seats, paying no attention to the seat assigned to them by their ticket, so exactly …416Let \(T\) be the centroid of an acute triangle \(ABC\). Let \(A'\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(A''\) be the point of the segment \(BC\) for which \[ BA' = A''C . \] …417Determine all natural numbers \(n\) with all of the following properties: \(n\) is divisible by \(2\) but not by \(4\); the sum of the digits of \(n\) equals \(6\); the number of divisors of \(n\) equals …418For a set \(X\) let \(\mathcal{P}(X) = \{Y : Y \subseteq X\}\) denote its power set. For example \(\mathcal{P}(\{1\}) = \{\varnothing, \{1\}\}\), since the subsets of \(\{1\}\) are \(\varnothing\) and …419Two players alternately write one of the numbers \[ 473, \quad 523, \quad 573, \quad 623, \quad 673, \quad 723, \quad 773, \quad 823, \quad 873 \] into a free cell of a \(3 \times 3\) table, where each …420A collection of \(2020\) consecutive positive integers is split into two subsets of \(1010\) numbers each. Can the least common multiple of all the numbers in the first subset be equal to the least common …421Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.422An \(8 \times 8\) board is tiled with copies of the three figures below. The figures may be rotated and reflected, and any number of copies of each of the three shapes may be used; the board counts as …423Let \(k \geq 2\) be a natural number. Margita has written on the board the first \(2k - 1\) natural numbers \(1, 2, \ldots, 2k - 1\). In one move she may erase any two numbers from the board and write …424Let \(A\) be a set of \(10\) numbers chosen from \(\{1, 2, \ldots, 100\}\). Prove that \(A\) has two nonempty subsets \(S\) and \(T\) with no element in common such that the sum of the elements of \(S\) …425Can the numbers \(1, 2, \dots, 100\) be distributed into three groups so that the sum of the numbers in the first group is divisible by \(102\), the sum of the numbers in the second group is divisible …426A polynomial \(p(x)\) has integer coefficients. Divided by \(x^2 - 12x + 11\), it leaves the remainder \(990x - 889\). Prove that no integer is a zero of \(p(x)\).427Let \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers \(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\) is divisible by \(5\). …428The smallest natural number whose square ends in three fours is \(38\), because \(38^2 = 1444\). Which natural number is the next smallest one with this property?429A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]430Is there a natural number \(n\) with the following property: if \(n\) is multiplied by the sum of its own digits, then the digits of the resulting product add up to \(3\)?431The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]432Let \(a\) and \(b\) be real numbers for which \[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \] Prove that \[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]433Find all real numbers \(x\) for which \[ \Biggl|\, \Bigl|\, \bigl|\, |x| - 2 \,\bigr| - 20 \,\Bigr| - 200 \,\Biggr| = 2007 . \]434Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.435Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]436The integers \(a\), \(b\), \(c\), \(d\) satisfy \(a > b > c > d\) and \[ (1 - a)(1 - b)(1 - c)(1 - d) = 10 . \] Which values can the expression \(a + b - c - d\) take?437Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.438Find every natural number \(n \ge 10\) all of whose decimal digits are nonzero and which has the following property: deleting any single digit of \(n\) leaves a number that divides \(n\).439Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]440Real numbers \(a\) and \(b\) satisfy \(|a| \ne |b|\) and \(a \ne 0\), together with \[ \frac{a-b}{a^{2}+ab} + \frac{a+b}{a^{2}-ab} = \frac{3a-b}{a^{2}-b^{2}} . \] Determine the value of \(\dfrac{b}{a}\). …441Eva, Igor, Marko and Marusa each wrote one natural number on a sheet of paper. Deleting the last digit of Eva's number gives Igor's number; deleting the last digit of Igor's number gives Marko's number; …442Jure drew four distinct lines in the plane, one arrangement after another, and for each arrangement he wrote down the number \(n\) of points at which at least two of his lines cross. Which of the sets …443In the convex quadrilateral \(ABCD\) the line through \(A\) and \(D\) and the line through \(B\) and \(C\) meet at a right angle, as the figure shows. Furthermore \(|CD| = 1\), \(|AC| = 2\) and \(|BD| = 3\). …444The horizontal segment in the picture is divided into six parts of equal length, and every triangle appearing in the picture is equilateral. The whole figure is shaded in two colours, light grey and dark …445What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)446Each of the five figures drawn below is bounded entirely by semicircular arcs, and the largest arc is the same in all five figures. Among the figures with the smallest perimeter, which one has the largest …447Peter covered a wound with two rectangular plasters, laid across each other as in the picture. The region covered by both plasters at once has area \(40 \ \mathrm{cm}^{2}\) and perimeter \(30 \ \mathrm{cm}\). …448In triangle \(ABC\) the angle at \(A\) is a right angle. Points \(D\), \(E\) and \(F\) are chosen on the sides \(AB\), \(BC\) and \(CA\) respectively, so that \[ |BD| = |BE| \qquad\text{and}\qquad |CF| = |CE| , \] …449Lara and Sara draw \(n\) straight lines on a rectangular sheet of paper, taking turns and drawing one line each time. Every line is parallel to one of the edges of the sheet and runs from edge to edge, …450On the sides of a triangle \(ABC\), equilateral triangles \(ADB\), \(BEC\) and \(CFA\) are constructed outwardly, so that \(D\), \(E\), \(F\) are the apexes over \(AB\), \(BC\), \(CA\) respectively. Prove …451An island is inhabited by \(45\) chameleons: \(17\) yellow, \(15\) grey and \(13\) blue. They wander about and meet from time to time, never more than two at a time. When two chameleons of the same colour …452Tine collects stamps. For his birthday he received a new album with room for plenty of them, so he took \(2002\) tolars out of his money box and decided to spend all of it on stamps. A friend offered him …453Oliver rolled an ordinary die \(100\) times and multiplied together all \(100\) numbers that came up on top. The product he obtained was \(6^{70}\). What is the smallest possible number of rolls on which …454Benjamin was working out the sum \(1 + 2 + 3 + \dots + 2012\). He left out a few of the terms, and the wrong total he ended up with was divisible by \(2011\). Anika was working out the sum \[ A = 1 + 2 + 3 + \dots + 2013 , \] …455Let \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …456On each of \(n > 4\) cards, one of the numbers \(+1\) and \(-1\) is written. A single question consists of naming exactly three of the cards, after which we are told the product of the numbers written …457The pupils of a school went to the theatre on two occasions, and every pupil of the school attended at least one of the two performances. Boys formed \(60\%\) of the audience at the first performance and …458Ali-Baba is standing in a cave full of gold and diamonds. A kilogram of gold is worth \(20\) dinars and a kilogram of diamonds is worth \(60\) dinars. He has a single chest with him. Filled with gold, …459Let \(a = 123456789\) and \(b = 987654321\). (1) Find \(\gcd(a, b)\). (2) Find the remainder left by \(\operatorname{lcm}(a, b)\) on division by \(11\).460Real numbers \(a, b, c, d\) satisfy \[ a^{2} + b^{2} = c^{2} + d^{2} , \qquad ab + cd > 0 , \qquad ac + bd > 0 . \] Prove that \(ad + bc > 0\).461The points of a plane \(\alpha\) are split between two nonempty sets \(A\) and \(B\): no point belongs to both, and every point belongs to one of them. Prove that some isosceles right triangle has all …462Let \(a\), \(b\), \(c\) be nonzero real numbers such that \[ a(b + c) + b(c + a) + c(a + b) = ab + bc + ca . \] Prove that the value of \[ \frac{a^{2}(b + c) + b^{2}(a + c) + c^{2}(a + b)}{abc} \] is an …463Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]464A large rectangle is cut into seven smaller rectangles, each of which has both of its side lengths equal to a whole number of metres. Five of the seven pieces have their areas written in them, as shown …465Let \(m\) be an arbitrary integer. Prove that there is at least one pair \((x, y)\) of integers for which \[ 2x^2 + 11xy + 12y^2 + 4x + 5y + 6 = 2m . \]466Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).467A circle can be inscribed in the trapezoid \(ABCD\), whose parallel sides are \(AB\) and \(CD\). Prove that the circle having \(BC\) as a diameter and the circle having \(AD\) as a diameter touch each …468The capital of a certain country is joined by a direct air route to each of the other \(2012\) cities. Moreover, every one of those \(2012\) cities is joined by an air route to at least one city other …469An operation \(\diamond\) on the set \(G = \{1, 2, 3, \dots, 2016\}\) is given by the table \[ \begin{array}{c|cccccc} \diamond & 1 & 2 & 3 & 4 & \cdots & 2016 \\ \hline 1 & 5 & 5 & 5 & 5 & \cdots & 5 \\ 2 & 1 & 2 & 5 & 5 & \cdots & 5 \\ 3 & 4 & 3 & 5 & 5 & \cdots & 5 \\ 4 & 5 & 5 & 5 & 5 & \cdots & 5 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 2016 & 5 & 5 & 5 & 5 & \cdots & 5 \end{array} \] …470Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …471Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] be a polynomial with integer coefficients. Suppose that \(P\) has two distinct integer zeros, neither of which is positive (\(P\) may have further zeros besides …472Find every three-digit number \(\overline{abc}\) whose digits \(a\), \(b\), \(c\) are all nonzero and which satisfies \[ \overline{abc} = 3 \cdot a! + 2 \cdot b! + c! . \] Here \(\overline{abc}\) is the …473Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …474Maksim and Mina play the following game. Maksim starts by drawing a line in the plane; Mina then draws a line different from it; Maksim then draws a line different from both lines already drawn, and so …475A point \(D\) is chosen on the side \(BC\) of a triangle \(ABC\). Points \(E\) and \(F\), both different from \(D\), are chosen on the line \(BC\) so that \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …476Two concentric circles \(k_1\) and \(k_2\) have radii \(a\) and \(b\). Consider all rectangles that have two vertices on \(k_1\) and the remaining two vertices on \(k_2\). Determine the rectangle of largest …477Let \(a\), \(b\), \(c\) be three distinct nonzero real numbers, and for real \(x, y \neq a\) set \[ V(x,y)=\frac{1}{(a-x)^{2}(a-y)^{2}}\Bigl((a-b)^{2}(c-x)(c-y)-(c-a)^{2}(b-x)(b-y)\Bigr). \] Prove that …478Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …479Let \(x\) and \(y\) be nonnegative real numbers with \(x + y = 2\). Prove that \[ x^2 y^2 \left( x^2 + y^2 \right) \leq 2 . \] When does equality hold?480How many integers \(n\) with \(10 \le n < 100000\) are divisible by \(4\), contain no digit \(0\) in their decimal representation, and have no two adjacent digits equal?481Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]482Tina wrote one natural number on each of five slips of paper and refused to say which numbers they were. Sharp-witted Zan talked her into revealing instead all the sums that can be formed from two of the …483A natural number is written on each face of a cube. At every vertex of the cube one writes the product of the numbers on the three faces that meet at that vertex. The eight numbers at the vertices add …484Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.485Determine every pair of coprime natural numbers \(m\) and \(n\) for which \[ \frac{5m - n}{m + n} \] is itself a natural number. (Here the natural numbers are the positive integers \(1, 2, 3, \ldots\).) …486Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).487Find the smallest natural number \(n\) that is divisible by \(20\) and for which \(n^2\) is a perfect cube and \(n^3\) is a perfect square.488The nonzero real numbers \(x\), \(y\), \(z\) satisfy \[ 3x + 2y = z \qquad \text{and} \qquad \frac{3}{x} + \frac{1}{y} = \frac{2}{z} . \] Prove that the value of \(5x^{2} - 4y^{2} - z^{2}\) is always an …489Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]490For a real number \(a\), let \([a]\) denote the largest integer that is not greater than \(a\). Find all integers \(y\) for which there exists a real number \(x\) satisfying \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …491Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).492The real numbers \(x\) and \(y\) satisfy \[ x^{3} + x^{2} + xy + x + y + 2 = 0 \qquad\text{and}\qquad y^{3} - y^{2} + 3y - x = 0 . \] Determine the value of \(x - y\).493A kangaroo called Pythagoras likes exactly those natural numbers that are divisible by \(4\), have digit sum \(3\), and have exactly five digits equal to \(0\) in their decimal representation. How many …494Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]495Find all natural numbers \(n\) whose cube equals the sum of the squares of three divisors of \(n\), where the three divisors need not be different from one another.496Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]497Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).498Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.499Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]500Find every real number \(x\) for which \[ \left(2x^{2} + 7x + 6\right)^{3} - \left(x^{2} + 3x + 2\right)^{3} = \left(x^{2} + 4x + 4\right)\left(117x^{2} + 128x + 52\right) . \]501A natural number \(n \geqslant 2\) is divided by each of the natural numbers \(1, 2, \ldots, n-1\) in turn, and all the remainders obtained are written down. Find every \(n\) for which the sum of the distinct …502How many functions \(f \colon \mathbb{R}^{+} \to \mathbb{R}\) are there such that \[ f\left(x + \frac{1}{x}\right) = x^{2} + \frac{1}{x^{2}} \] holds for every \(x \in \mathbb{R}^{+}\)?503For each \(n = 1, 2, 3, \ldots\) Perica looks for the smallest block of \(2n+1\) consecutive positive integers with the property that the sum of the squares of the smallest \(n+1\) of them equals the sum …504Two vertical poles stand on level ground, at a distance of \(9\) m from each other; one pole is \(11\) m high and the other is \(15\) m high. A rope of length \(15\) m is fastened to the top of one pole …505In the expression \[ *\,1 * 3 * 3^{2} * 3^{3} * \cdots * 3^{1997} * 3^{1998} \] Arkadije and Branislav take turns replacing one of the stars by \(+\) or by \(-\), one star per move, until no star is left. …506Let \(ABCD\) be a parallelogram whose interior angle at \(A\) is acute, and let \(E\) be a point of the plane such that \(EA \perp AB\) and \(EC \perp CB\). Prove that \[ \angle AED = \angle CEB . \]507Let \(ABC\) be a triangle and let \(M\), \(N\), \(P\) be points on its sides \(AB\), \(BC\), \(AC\) respectively, chosen so that \(AMNP\) is a parallelogram. Let \(k_1\) be the circle circumscribed about …508In a group of \(20\) people, every person chooses ten of the other nineteen and sends one letter to each of them. Prove that there are two people who sent a letter to each other.509How many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …510Let \(E\) be a point on the side \(CD\) of a square \(ABCD\). The point \(F\) lies on the line \(AB\) but not on the segment \(AB\), and satisfies \(|BF| = |DE|\). Prove that the lines \(AC\) and \(EF\) …511Sixteen points of the integer lattice are marked, as in the picture: all points \((x,y)\) with \(x\) and \(y\) taken from \(\{1,2,3,4\}\). At most how many of these points can be coloured red so that no …512For a natural number \(n\), let \(S(n)\) be the sum of its decimal digits and \(P(n)\) the product of its decimal digits. Find all natural numbers \(n\) for which \[ S(n) + P(n) = n . \]513Let \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which \[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \] …514Find every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]515Ana picked the eight digits \(1, 2, 3, 4, 5, 6, 7\) and \(9\). She then forms groups of four two-digit primes, each group using all of her chosen digits. What is the sum of the four primes in one such …516In triangle \(ABC\) the side \(AB\) is twice as long as \(AC\), that is \(|AB| = 2|AC|\). The point \(D\) lies on the ray \(CA\) and satisfies \(|CD| = 3|AC|\). Prove that the perpendicular dropped from …517Prove that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).518On \(41\) squares of a chessboard - the ordinary \(8 \times 8\) board - a king is placed, one king on each of those squares. Prove that among these kings one can find three pairwise disjoint sets, each …519Two circles \(R_1\) and \(R_2\) meet at points \(A\) and \(B\). A line through \(A\) is allowed to vary; it meets \(R_1\) again at \(P\) and \(R_2\) again at \(Q\). Prove that all the resulting perpendicular …520A circle \(k\) has radius \(31\,\mathrm{mm}\), and \(\ell\) is a broken line of length \(61\,\mathrm{mm}\) whose two endpoints both lie on \(k\). Prove that there is a line \(p\) passing through the centre …521Let \(ABCD\) be a trapezoid with \(AB \parallel CD\), and let \(P\) be a point on the extension of the diagonal \(AC\) beyond \(C\), so that \(C\) lies between \(A\) and \(P\). Let \(X\) and \(Y\) be the …522For a natural number \(n\), let \(x_n\) be the number obtained by writing the natural numbers from \(1\) to \(n\) one after another, for example \[ x_{15} = 123456789101112131415 . \] Find all natural …523Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?524Let \(n > 1\) be a natural number. How many \(n\)-digit numbers are palindromes and divisible by \(9\)? (A number is a palindrome when its decimal representation is symmetric, that is, it reads the same …525Let \(ABC\) be an acute triangle and let \(D\) be the foot of the altitude from \(A\), so that \(D\) lies on the side \(BC\). A point \(P\) is chosen on the segment \(AD\) in such a way that \[ \angle PBA = \angle PCA . \] …526Let \(k > 0\). On the sides \(A_1B_1\), \(B_1C_1\) and \(C_1A_1\) of a triangle \(A_1B_1C_1\), points \(C_2\), \(A_2\) and \(B_2\) are chosen, respectively, so that \[ \frac{A_1C_2}{C_2B_1} = \frac{B_1A_2}{A_2C_1} = \frac{C_1B_2}{B_2A_1} = k . \] …527Let \(a\), \(b\), \(c\) be the side lengths of a triangle \(ABC\), let \(S\) be its area and \(R\) the radius of its circumscribed circle, and let \(M\) be a point in the interior of the triangle. Write …528Let \(a\), \(b\), \(c\) be arbitrary positive integers. Prove the inequality \[ \gcd(a,\,b-1)\cdot\gcd(b,\,c-1)\cdot\gcd(c,\,a-1) \;\le\; ab+bc+ca-a-b-c+1 , \] and prove that equality is attained for infinitely …529In every cell of a table with \(2017\) rows and \(2017\) columns one of the numbers \(1, 2, 3, \dots, 2017\) is written. Is it possible to do this so that in every row, in every column and along every …530For a positive integer \(n\), let \(x_n\) be the number obtained by writing the decimal representations of \(1, 2, \dots, n\) one after another; for example \(x_{14} = 1234567891011121314\). Define \(f \colon \mathbb{N} \to \mathbb{N}_0\) …531Prove that there exist infinitely many pairs \((m, n)\) of distinct positive integers such that the sum of all positive divisors of \(m^2\) is equal to the sum of all positive divisors of \(n^2\).532Positive real numbers \(a, b, c, d, e\) satisfy \[ a(b+c) = b(c+d) = c(d+e) = d(e+a) = e(a+b) . \] Prove that \(a = b = c = d = e\).533A square board of size \(n \times n\) is given, where \(n \geq 2\). The numbers \(1, 2, \dots, n^2\) are written into the \(n^2\) unit cells of the board, one number in each cell, each number used exactly …534Decide whether there exists a polynomial \(P(x)\) with integer coefficients such that, for some four pairwise distinct integers \(a\), \(b\), \(c\), \(d\), \[ P(a) = P(b) = P(c) = P(d) = 2024 , \] and …535Let \(n\) be a natural number and let \(d\) be a positive divisor of \(2n^2\). Can \(n^2 + d\) be a perfect square?536A triangle \(ABC\) is given. Find every point \(M\) of its plane for which the three triangles \(ABM\), \(BCM\) and \(CAM\) have equal areas.537Let \(k > 3\) and consider the number \(2^k\). Prove that no rearrangement of the decimal digits of \(2^k\) can produce the number \(2^n\) with \(n > k\).538The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …539Let \(AB\) be a segment and let \(P\) be any point of it other than \(A\) and \(B\). On the hypotenuses \(AP\) and \(PB\) erect isosceles right triangles \(APQ\) and \(PBR\), with the right angles at the …540A circle \(k\) has diameter \(AB\). A point \(M\) of \(k\) is chosen, different from \(A\) and from \(B\). Let \(k_1\) be the circle with centre \(M\) that touches the diameter \(AB\). The line \(AB\) …541Let \(E\) and \(F\) be the midpoints of the sides \(AD\) and \(DC\) of a rectangle \(ABCD\), and let \(G\) be the point where the segments \(AF\) and \(EC\) cross. Prove that \[ \angle CGF = \angle FBE . \] …542The integers \(x\) and \(y\) satisfy \[ x + xy + y^{2} = 1 \qquad\text{and}\qquad y(5 + x) \ge 0 . \] Which integer values can the expression \(x - y\) take?543A square \(ABCD\) is given, together with points \(E\) and \(F\) lying outside the square such that the triangles \(BEC\) and \(CFD\) are equilateral. Prove that the triangle \(AEF\) is equilateral as …544Find all pairs of coprime integers \(x\) and \(y\) that satisfy the equation \[ 4x^{3} + y^{3} = 3xy^{2} . \]545Let \(ABC\) be a triangle. Points \(D\) and \(E\) lie on the rays \(CA\) and \(CB\) respectively, but not on the sides of the triangle \(ABC\), and are chosen so that \[ |AD| = |BE| = |AB| . \] Let \(G\) …546Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \frac{3xy}{x-y} = 2, \qquad \frac{2yz}{y+2z} = 3, \qquad \frac{xz}{z-4x} = 3 . \]547A disc \(K\) of radius \(R\) is cut into three circular sectors in such a way that the areas of the two smaller sectors add up to the area of the largest sector, while the difference of the areas of the …548In how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …549Two fixed points \(A\) and \(B\) are given in the plane. A point \(M\) is chosen and then travels along the straight segment from \(M\) to \(A\). Determine all positions of \(M\) for which the distance …550Stars are drawn in the cells of a \(4 \times 4\) table, at most one star per cell. What is the least number of stars for which the following holds: whichever \(2\) rows and whichever \(2\) columns are …551Ales, Brane and Cvetka made a large pile of cards, writing on each card one of the numbers \(2, 3, 4, 5, 6, 7, 8\); every one of these numbers appears on many cards. Maja, who arrived later, picked three …552In a triangle \(ABC\) we have \(\angle ABC = 45^\circ\) and \(\angle CAB = 15^\circ\). Let \(M\) be the point of the ray \(BC\) for which \[ \overrightarrow{BM} = 3 \cdot \overrightarrow{BC} . \] Determine …553Call a positive integer symmetric if its decimal representation reads the same from left to right as from right to left. Prove that there are infinitely many positive integers \(n\) for which the numbers …554Circles \(k_1, k_2, \ldots, k_{1999}\) lie in the plane and touch one another externally in a closed chain: \(k_1\) touches \(k_2\) at \(A_1\), \(k_2\) touches \(k_3\) at \(A_2\), and so on, and finally …555A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …556A pentagon \(ABCDE\) is inscribed in a circle of radius \(r\), and three of its sides have length \(r\): \[ AB = BC = DE = r . \] Let \(G\) and \(F\) be the midpoints of the sides \(CD\) and \(EA\). Prove …557Eighteen matches are laid out to form the grid shown below: an equilateral triangle whose side is three matches long, divided into nine small triangles. What is the smallest number of matches that must …558Three squares are inscribed in a right triangle \(ABC\) with the right angle at \(B\), arranged as in the picture: the largest one has two sides lying on the legs of the triangle, and each of the two smaller …559Let \(A\) be the set of all integers from \(-20\) to \(20\), that is \[ A = \{\, a \in \mathbb{Z} \;:\; -20 \le a \le 20 \,\} . \] Let \(n\) be a positive integer and let \(A_1, A_2, \ldots, A_n\) be pairwise …560A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way …561Let \(ABCDEF\) be a regular hexagon. A point \(M\) is taken on the diagonal \(AC\) and a point \(N\) on the diagonal \(CE\) so that both are placed at the same relative position: \[ \frac{AM}{AC} = \frac{CN}{CE} = \lambda . \] …562Find the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …563Let \(P(x)\) be a polynomial with integer coefficients for which there exist prime numbers \(p < q < r\) with \[ \{P(p),\, P(q),\, P(r)\} = \{20,\, 3,\, 2010\} . \] Prove that \(P(p+q) = 2010\).564It is known that for some positive integers \(x\) and \(y\), \[ 23^{x} \cdot 111^{y} = \overline{aab3dc6902b2c74d456b} , \] where \(a, b, c, d\) are digits, not necessarily different, and \(a \neq 0\). …565Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).566The numbers \(a, b, c, x, y, z\) satisfy \[ \{a, b, c\} = \{x, y, z\} = \{15, 3, 2014\} . \] Must the number \[ a^{b^{c}} + x^{y^{z}} \] be composite? (For \(m, n, k \in \mathbb{N}\), the symbol \(m^{n^{k}}\) …567Determine all polynomials \(R(x)\) whose coefficients all belong to the set \(\{-1, 1\}\) and which satisfy \[ R(3) = 130 \qquad \text{and} \qquad R(-2) = -45 . \]568Let \(ABC\) be an acute triangle with \(AB < AC\), and let \(D\) be the midpoint of its side \(BC\). Let \(p\) be the image of the line \(AD\) under reflection in the bisector of the angle \(BAC\), and …569Determine all pairs of positive integers \(a\) and \(b\) for which the number \[ a^4 b + 3b - 2a^2 b^2 - a^2 - 3b^3 \] is a power of two.570Let \(k\) be a positive integer and let \(A\), \(B\), \(C\) be sets such that \(|A \triangle B| = |B \triangle C| = |C \triangle A| = 2k .\) Prove that there is exactly one set \(D\) for which \(|A \triangle D| = |B \triangle D| = |C \triangle D| = k .\) …571Two players play the following game. Taking turns, each player writes down one digit, the digits appearing in a row from left to right in the order in which they are written, and no player is allowed to …572Determine every natural number \(n\) for which there exist real numbers \(a\), \(b\), \(c\) satisfying \[ \bigl\{\, a + b + c,\; ab + bc + ca,\; abc \,\bigr\} = \{\, n,\; n+1,\; n+2 \,\} . \]573Let \(ABC\) be a triangle with \(\angle CAB = 60^\circ\). Denote by \(O\) and \(I\) the centres of the circle circumscribed about it and of the circle inscribed in it, respectively, and let \(A'\) be the …574A point \(P\) inside a triangle \(ABC\) satisfies \[ \angle BPC = \angle BAC + 60^\circ, \qquad \angle CPA = \angle CBA + 60^\circ, \qquad \angle APB = \angle ACB + 60^\circ. \] The lines \(AP\), \(BP\), …575An octagon has all of its interior angles equal, and the lengths of all of its sides are rational numbers. Prove that it has a centre of symmetry.576A triangle \(ABC\) is given. Consider all lines which cut the side \(AC\) at a point \(M\) and the side \(BC\) at a point \(N\) in such a way that \(MN = AM + BN\). Prove that there is a circle \(k\) which …577On the sides \(BC\), \(CA\) and \(AB\) of a triangle \(ABC\) points \(A_1\), \(B_1\) and \(C_1\) are marked, respectively. Let \(T\) be the centroid of the triangle \(ABC\) and \(T_1\) the centroid of …578How many triples \((a, b, c)\) of positive integers are there such that \(2a + 1\) is divisible by \(b\), \(2b + 1\) is divisible by \(c\), and \(2c + 1\) is divisible by \(a\)?579Let \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(D\) be the point of the side \(AC\) for which \(CD = 2 \, AD\), and let \(P\) be a point of the segment \(BD\) with \(\angle APC = 90^\circ\). …580Andraz and Breda cut two long strips out of a newspaper, of lengths \(a\) and \(b\), to play a game with. A move consists of choosing one of the strips and cutting a piece of length \(d\) off it, so that …581A spider has spun the web shown below: five regular octagons nested one inside the other, with each vertex of an octagon joined by a thread to the corresponding vertex of the neighbouring octagons. The …582Find the smallest natural number \(n\) for which an \(n \times n\) board of unit cells can be covered completely and without overlaps by equally many tiles of the two shapes below: an \(L\)-shaped tile …583The triangle \(ABC\) has side lengths \(|AB| = 15\) cm, \(|BC| = 14\) cm and \(|CA| = 13\) cm. Let \(D\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(E\) be the point of that …584Let \(ABCD\) be a convex quadrilateral and let \(E\) and \(F\) be points on the sides \(AB\) and \(AD\) respectively, chosen so that \(EF \parallel BD\). The segment \(CE\) meets the diagonal \(BD\) at …585In a rectangle \(ABCD\) with \(|AB| > |BC|\), the perpendicular bisector of the diagonal \(AC\) meets the side \(CD\) at the point \(E\). The circle with center \(E\) and radius \(|AE|\) meets the side …586A \(4 \times 4\) table is divided into \(16\) unit cells. On this table we place tiles of the shape drawn alongside: two unit squares that meet at a single corner. A tile may be rotated, and each tile …587A triangle \(ABC\) has a point \(D\) on side \(AB\) and a point \(E\) on side \(AC\) such that \[ |AE| = |ED| = |DB| \qquad\text{and}\qquad |AD| = |DC| = |CB| . \] Determine the angles of triangle \(ABC\). …588A parallelogram \(ABCD\) satisfies \(|AB| = |BD|\). Let \(K\) be the point of line \(AB\), different from \(A\), with \(|KD| = |AD|\). Let \(M\) be the image of \(C\) under the half-turn about \(K\) (so …589Let \(n \ge 2\) be a natural number. Maja and Peter want to colour every cell of an \(n \times n\) table either black or blue, subject to one rule: among any four cells that can be covered by a square …590The lines containing the two diagonals of a quadrilateral \(ABCD\) meet at an angle of \(60^{\circ}\). Each vertex of the quadrilateral is reflected in the line containing the diagonal that joins its two …591Natasa glued a square and an equilateral triangle of the same side length into a pentagon. Out of seven copies of that pentagon she built the figure shown on the right, which sits inside a large square. …592Three curves are drawn inside a square \(ABCD\): the quarter circle \(\mathcal{Q}\) centred at the vertex \(A\) and passing through \(B\) and \(D\); the semicircle \(\mathcal{P}\) centred at the midpoint …593Determine all quintuples of primes \(p_{1} \le p_{2} \le p_{3} \le p_{4} \le p_{5}\) with the property that each of the five primes divides the sum of the remaining four.594The figure below consists of a triangle \(ABC\) together with the circular segment erected on the side \(BC\), on the opposite side of \(BC\) from \(A\). Construct at least one straight line that divides …595Let \(H\) and \(O\) be the orthocentre and the circumcentre of a triangle \(ABC\) with \(AB \neq AC\). The lines \(AH\) and \(AO\) meet the circumcircle of \(ABC\) a second time at \(M\) and \(N\) respectively. …596Let \(P(x)\) be a polynomial with integer coefficients such that, for every positive integer \(n\), dividing \(P(P(n))\) by \(n\) leaves remainder \(n - 1\). Prove that \(P(x)\) has no integer root.597Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(abcd = 1\) and \[ a + b + c + d = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} . \] Prove that some two of the numbers \(ab\), \(ac\), …598An acute triangle \(ABC\) is given. Construct, with ruler and compass, a point \(P\) inside the triangle such that the rays \(AP\), \(BP\) and \(CP\) meet the circumcircle of \(ABC\) again in the three …599Let \(ABC\) be an acute triangle. The bisector of its interior angle at \(B\) meets \(AC\) at \(K\), and \(CD\) is the altitude from \(C\), with \(D\) on \(AB\). Let \(N\) be the point of \(CD\) for which …600Let \(M\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\), so that \(AB \parallel CD\). A point \(E\) is taken in the interior of the segment \(AC\) in such a way that the lines \(BC\) and \(ME\) …601Let \(I\) be the incenter of a triangle \(ABC\), and suppose that \[ |CA| + |AI| = |BC| . \] Determine the ratio of the sizes of the angles \(\angle BAC\) and \(\angle CBA\).602For some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?603Level 1 of the computer game Zakladnica takes place in an underground treasury built from \(13\) octagonal and \(12\) square rooms, arranged as in the figure. The only way into the treasury, and the only …604Three points \(A\), \(B\), \(C\), not lying on one line, are given. Construct a point \(D\) for which the quadrilateral \(ABCD\) is at the same time cyclic and tangential, that is, admits both a circumscribed …605In the plane of a triangle \(ABC\) one draws \(n\) lines, each of them parallel to one of the three sides of the triangle. Determine the smallest \(n\) for which these \(n\) lines can cut the plane into …606In a triangle \(ABC\) the angle at \(B\) equals \(80^\circ\). Three further points are marked: the point \(D\) on the side \(BC\) with \(AB = AD = CD\); the point \(F\) on the side \(AB\) with \(AF = BD\); …607Can nine points, no three of them collinear, be placed inside the cross-shaped figure below (its boundary included) in such a way that whenever three of them span a triangle lying inside the figure, that …608Let \(ABC\) be an isosceles triangle with \(AB = AC\). A point \(P\) is taken inside the triangle so that \[ \angle BPC = 90^\circ + \tfrac{1}{2}\angle BAC , \] and a point \(Q\) is taken so that \(\angle BPQ = \angle PQA = 90^\circ\). …609Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that \[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \] …610The product of the binomial coefficient \(\binom{64}{21}\) and an unknown odd number equals \[ 5{\ast}\,6{\ast}0\,{\ast}8{\ast}\,862\,{\ast}1{\ast}\,7{\ast}7\,{\ast}4{\ast}\,4{\ast}5\,12{\ast}\,9{\ast}{\ast} \, , \] …611Let \(ABC\) be a triangle. Prove that the following three lines all pass through one point: the bisector of the angle at \(A\); the line through the midpoints of the sides \(CA\) and \(CB\); and the line …612Prove that the disk of radius \(100\) centred at the origin contains fewer than \(31600\) points whose two coordinates are both integers. (A point counts as contained in the disk if it lies inside it or …613Let \(n\) be a natural number divisible by \(6\). Prove that there exists a convex \(n\)-gon whose interior angles are all equal and which can be cut into finitely many pieces, each piece being of one …614All powers of two are written on a board in increasing order: \(1, 2, 4, \ldots\). Aca and Braca now take turns, Aca first. A move consists of choosing two numbers that stand next to each other on the …615For a positive integer \(n\), let \(f(n)\) denote the least common multiple of the numbers \(1, 2, \ldots, n\). Find all positive integers \(n\) for which \[ f(n) < f(n+1) < f(n+2) < f(n+3) . \]616Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy \[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \] …617Let \(ABC\) be a triangle and let \(k\) be its circumcircle, with centre \(O\). Construct a point \(D\) on \(k\) such that the centroids of the triangles \(ABC\) and \(ABD\) are collinear with the point …618For a positive integer \(x\), let \(S(x)\) denote the sum of the decimal digits of \(x\). (a) Determine the smallest element of the set \(\{\, S(11n^2 + n + 1) \mid n \text{ a positive integer} \,\}\). …619The circle inscribed in a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(M\) and \(N\) respectively. Let \(P\) be the point in which the bisector of the angle \(ABC\) meets the line …620In an equilateral triangle \(ABC\) the side has length \(|AB| = 2\). Let \(M\) and \(N\) be interior points of the side \(AB\) with \(|MN| = 1\). Prove that \[ \angle MCN > 30^\circ . \]621A mole has dug a number of underground rooms and joined them by tunnels, in such a way that from every room exactly \(3\) tunnels lead out, to \(3\) different rooms. Tunnels meet one another only at rooms. …622Let \(I\) be the centre of the inscribed circle of a triangle \(ABC\), and let \(A_1\), \(B_1\), \(C_1\) be the feet of the perpendiculars dropped from \(I\) to the sides \(BC\), \(AC\) and \(AB\). The …623Let \(D\) be the midpoint of side \(AB\) of an acute triangle \(ABC\). Points \(A'\) and \(B'\) are chosen on the segments \(AC\) and \(BC\) so that the triangles \(ADA'\) and \(DBB'\) are both isosceles …624A natural number is written in every cell of a square table. Call the table interesting if the sum of all the numbers in it is odd and, in addition, the sum of the four numbers covered by any placement …625A teacher handed Matej four sheets of paper, each carrying one nonzero digit. Matej laid the sheets in a row and so formed a four-digit number. He then interchanged two of the sheets, without flipping …626In a hexagon \(ABCDEF\) the following hold: \(\angle BAF = 150^{\circ}\), \(\angle ACB = \angle ADC = 90^{\circ}\), \(|AC| = |BC|\), triangle \(ABC\) is similar to triangle \(ADE\), and triangle \(BCD\) …627We want to cover a \(4 \times 4\) board with tiles of the shape shown below, rotations and reflections being allowed. The tiles are permitted to overlap one another and to stick out beyond the edge of …628Three piles of tokens lie on a table, holding \(a\), \(b\) and \(c\) tokens, where \(a \ge b \ge c > 0\). Players \(A\) and \(B\) move tokens alternately, and \(A\) starts. In one move a player first selects …629A rectangular grid of size \(7 \times 9\) is given: seven rows of cells and nine columns of cells, as in the figure. At the bottom-left node of the grid sits a colony of ants, and at the top-right node …630Find all pairs of real numbers \(x\) and \(y\) that satisfy the system \[ \frac{y^{3} + 15x^{2}}{y^{4} - x^{3}} = \frac{y^{2} + 15x}{y^{3} - x^{2}} , \] \[ \frac{1500y^{3} + 4x^{2}}{9y^{4} - 4} = \frac{1500y^{2} + 4x}{9y^{3} - 4} . \] …631Timotej had a sheet of squared paper measuring \(8 \times 8\) little squares. He folded it a few times, each fold running along one of the lines of the grid, until he was left with a square piece measuring …632Finitely many arcs are marked on a circle. The length of each of them is smaller than half of the circumference, and any three of the marked arcs have a common point. Prove that there is a point of the …633A strip of \(1 \times n\) cells is given, where \(n > 10\) is a natural number, and its cells are numbered \(1, 2, \dots, n\) from left to right. Cell number \(10\) is black and carries a token; every …634We want to choose a set \(P\) of \(k\) primes and a set \(N\) of \(n\) consecutive positive integers in such a way that every number \(a \in N\) is divisible by at least one prime \(p \in P\). (a) Determine …635Anja owns tiles shaped like a single unit square, Bojan tiles shaped like an L-tromino: three unit squares forming an L, as drawn below. The two players alternately place one tile of their own onto a rectangular …636Every cell of an \(n \times n\) table contains the number \(0\). One step consists of choosing three cells that form the shape and adding \(1\) to each of the three numbers standing in them. Can we, after …637Let \(C\) be a point of the segment \(AB\) other than \(A\) and \(B\), and let \(k_{0}, k_{01}, k_{02}\) be the circles with diameters \(AB, AC\) and \(CB\) respectively. Let \(D\) be a point in which …638For which positive integers \(m\) and \(n\) can an \(m \times n\) rectangle be covered completely and without overlaps by copies of the three figures shown below, each of them built from unit squares? …639A pile of \(n\) tokens lies on a table. Two players, \(A\) and \(B\), move alternately, and \(A\) moves first. In one move a player must do one of the following: remove one token from one of the piles …640Let \(ABC\) be a right triangle with hypotenuse \(AB\), and let \(D\) be the midpoint of \(AB\). Let \(k\) be the circumcircle of the triangle \(BCD\), and let \(E\) be an arbitrary point of the shorter …641In a triangle \(ABC\), let \(O\) and \(I\) be the centres of the circumscribed and of the inscribed circle, respectively. Let \(O_1\) be the image of \(O\) under the reflection in the point \(I\), so that …642Determine all nonnegative integers \(n\) and all digits \(a\), \(b\), \(c\) for which the number \[ M = \overline{1\,\underbrace{0 \dots 0}_{n}\,a\,\underbrace{0 \dots 0}_{n}\,b\,\underbrace{0 \dots 0}_{n}\,c} \] …643Determine the largest possible value of \(n\) for which there exists a convex \(n\)-gon that can be decomposed into a disjoint union of triangles, each of which is either right isosceles or right-angled …644Let \(ABC\) be a triangle, let \(D\) be the foot of the altitude from \(A\) to the line \(BC\), and let \(\omega\) be the circle whose diameter is the segment \(AD\). Denote by \(G\) the second common …645A finite sequence of \(n\) numbers is given, each of them equal to \(0\) or \(1\), where \(n\) is a positive integer. One move consists of choosing two adjacent terms \(x\) and \(y\), deleting both of …646Circumscribe about a given triangle \(ABC\) an equilateral triangle \(PQR\) whose side is as long as possible. (Here \(\triangle PQR\) is called circumscribed about \(\triangle ABC\) when \(A \in QR\), …647A bank guard looks after \(n\) safes. Every safe has its own key, no key fits two safes, and all the keys look alike. He is given as many identical circular metal rings as he wants. On any ring he may …648Every point of three-dimensional space is coloured with one of two colours, red or blue, in such a way that whenever three points \(A\), \(B\), \(C\) have the same colour and \(AB = AC\), the midpoint …649Every point of space is painted in one of three colours. Prove that one of the three colours can be chosen in such a way that for every positive real number \(r\) there exists a triangle of area \(r\) …650Let \(n\) be a positive integer. What is the largest number of rooks that can be placed on an \(n \times n\) board so that every rook attacks at most \(3\) of the other rooks? Attacks are the usual chess …651Determine the smallest positive integer \(n\) with the following property: for some real numbers \(a_0, a_1, \dots, a_n\) the function \(f : \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \bigl| \, \cdots \, \bigl| \bigl| \, |x - a_0| - a_1 \bigr| - a_2 \bigr| - \cdots - a_{n-1} \bigr| - a_n, \qquad x \in \mathbb{R}, \] …
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