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 …171Prove 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 …172Let \(S\) 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 …173Each 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?174A 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 …175Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).176Let \(ABC\) 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 \(A_2\) …177Solve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.178Baron 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 …179In 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. …180Is it possible to divide a square into convex pentagons?181The 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 …182In 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 …183Let \(X\) 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 \(\angle ADX\), \(\angle DXC\) …184Find all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).185Jure 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 …186Find all real numbers \(x\) that satisfy the inequality \[ \frac{|x-3| + x}{x+1} < 1 . \]187Find all pairs of natural numbers \(a\) and \(b\) that satisfy the equation \[ a^2 - 5ab + 24 = 0 . \]188Suppose the numbers \(a\) and \(b\) satisfy \(a + b = 1\) and \(ab \neq 0\). Prove that \[ \frac{a}{b^3 - 1} - \frac{b}{a^3 - 1} = \frac{2(b-a)}{a^2b^2 + 3} . \]189Let \(M\) 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 …190Nika 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 …191Andrej 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 …192Let \(a\) 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.193The 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 …194A 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 \(CDM\), …195A 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 …196Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]197Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).198Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.199Let \(a\) 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 \(a + b\)?200The real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).201In 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 \[ \angle BDC = 90^\circ \qquad\text{and}\qquad \angle ACD = \angle DBA . \] …202Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]203Grandmother 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, …204For 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 \((x, y)\)?205The 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 …206Let \(ABCD\) 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 …207In 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 …208Find all pairs of real numbers \(x\), \(y\) that satisfy the system \[ \begin{aligned} \frac{3}{x-4y} + \frac{2}{x+y-5} &= 0, \\ \frac{2}{x^{2}-4y^{2}} + \frac{1}{x^{2}+y^{2}-5} &= 0. \end{aligned} \]209Alenka 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 …210In 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 …211Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]212Let \(ABC\) 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 …213Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.214The 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 \(G\) …215The 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.216Farmer 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 …217Find 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.218Vid 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 …219At 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 …
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