Practice library
Problems
1Find 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.Open2An angle of \(7^\circ\) is given. Using only compass and straightedge, divide it into seven equal parts.3Determine the smallest natural number the product of whose digits equals \(75600\).4Write 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 …5Into 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 …6In 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 …7In 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 …8The 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.9Let \(a\), \(b\), \(c\) 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 \(p\), \(q\), \(r\) …10Find all prime numbers \(p\), \(q\), \(r\), not necessarily different from one another, and all positive integers \(n\), for which \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{n}. \]11A 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 \(A\) …12Natural numbers \(a\), \(b\) and \(c\) satisfy \[ a + \cfrac{1}{b + \cfrac{1}{c}} = \frac{4016}{2007} . \] Prove that \[ \cfrac{1}{c + \cfrac{1}{b + \cfrac{1}{a}}} = \frac{2007}{4016} . \]13It 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) …14Does 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 \(x\)?15Does the number \[ 2010^{2010} + 10^{2011} \] have more digits in its decimal representation than the number \(2010^{2010}\)?16In how many ways can \(11\) birds be placed into \(3\) identical cages so that every cage contains at least three birds?17For a natural number \(k\), let \(S(k)\) denote the sum of its digits. Do there exist natural numbers \(n\) and \(m\) such that \[ S(n) \cdot S(n+1) \cdot \ldots \cdot S(n+m) = 2011^{2010}\,? \]18Determine 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 \(n\).19Ten 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, …20At 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, …21Determine 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 \(1, 2, \dots, 2017\) …22Let \(K\) 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 …23Find 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 …24Prove 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.25Three 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 …26A 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 …27Let \(A\) and \(B\) be non-empty sets, neither of which is a subset of the other. For a natural number \(n\) consider the equality \[ \underbrace{A \setminus \bigl(B \setminus (A \setminus (B \setminus \cdots))\bigr)}_{n \text{ sets}} \;=\; \underbrace{A \mathbin{\triangle} \bigl(B \mathbin{\triangle} (A \mathbin{\triangle} (B \mathbin{\triangle} \cdots))\bigr)}_{n \text{ sets}} \] …28Let \(xOy\) 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 …29Determine all pairs of prime numbers \(p\) and \(q\) for which \((p^3 + 1)^q\) is the square of a natural number.30Aca 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 …31Two 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 …32Let \(A\) 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 \(2000\).33Let \(a\) 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 …34Find a five-digit natural number whose half is the square of a natural number and whose third is the cube of a natural number.35Prove 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.36The 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 …37A 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 …38Let \(S = \{1, 2, \ldots, 20\}\). What is the largest possible number of elements of a subset \(A \subseteq S\) with the property that \(2x \notin A\) whenever \(x \in A\)?39Let \(a\), \(b\) 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 \(|p - q| < 1\).40Let \(n\) be a natural number. Prove that \(n^2 + 3n + 5\) is never divisible by \(121\).41a) 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 …42Find all pairs of real numbers \((x, y)\) satisfying \[ \frac{|x+y|}{1+|x+y|} = \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}. \]43Determine 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.44In 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 \(BE \parallel FA\).45Every 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 …46Two 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 …47In 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 \(2000\)?48In 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?49On 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 \(B_1C_1\). …50Does 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 …51Several 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 …52Find all values \(a \in \mathbb{R}\) for which the equation \[ |x - a| + |a - 1| = 1 \] has two solutions, and determine those solutions.53Words 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 …54In 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 …55In 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.56There 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?57Let \(CD\) 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.58A 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.59Let \(M\) 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 …60In 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 \[ 2\cdot\angle ACB > \angle BAC. \] …61Let \(ABCD\) be a quadrilateral such that \[ \angle BCA + \angle CAD = 180^{\circ} \qquad\text{and}\qquad AB = AD + BC. \] Prove that \(\angle BAC + \angle ACD = \angle CDA\).62Consider 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, \[ a_{i+1} - a_i \in \{2, 4\} \quad \text{for every } i. \] …63Let \(ABCDE\) 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.64Let \(A\), \(B\), \(C\), \(D\) 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 …65A 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 …66Find all natural numbers \(n\) for which the three numbers \[ n-4, \qquad 2n+2, \qquad 4n+1 \] are all perfect cubes.67Find 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 …68Let \(m > 1\) 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 …69Let \(n \ge 3\), 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 \[ \overline{b_1b_2\ldots b_n}, \] …70The 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 \[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \] …71Two 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 \(PB\) …72The 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, …73The 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 …74The 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 …75Two 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 …76A 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 …77For 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 …78Let \(E\) 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 …79Determine all triples of integers \((x, y, z)\) that satisfy \[ x^{2} + y^{2} + z^{2} = 2004\,xyz . \]80Can an equilateral triangle be divided - that is, actually cut up with scissors - into \(2006\) equilateral triangles?81One 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 …82What 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?83A 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 …84Let \(ABC\) 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 …85Ten 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 …86Two circles that do not intersect are given. Construct all of their common tangent lines.87Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]88Find 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 \[ Q(x) = x^{2} - (a+1)x + a. \]89In 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 …90A 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, …91On 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) …92Prove 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 …93Let \(S\) 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 …94Each 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?95A 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 …96Find all solutions of the equation \[ 6\left(6a^{2} + 3b^{2} + c^{2}\right) = 5d^{2} \] in integers \(a\), \(b\), \(c\), \(d\).97Let \(ABC\) 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 \(A_2\) …98Solve the equation \[ 12^{x} + 10^{y} = 7102^{z} \] in the set of natural numbers.99Baron 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 …100In 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. …101Is it possible to divide a square into convex pentagons?102The 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 …103In 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 …104Let \(X\) 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 \(\angle ADX\), \(\angle DXC\) …105Find all natural numbers that are powers of \(3\) and whose representation in base \(12\) contains only the digits \(6\) and \(9\).106Prove 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.107Find 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 \(1\).108Let \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]109Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).110The 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 …111Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).112The 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 …113Let \(p\), \(q\), \(r\) 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 \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …114Let \(x\), \(y\) 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 \(x + y + z\).115Aca, 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 …116Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]117Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).118Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …119How 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 \(3\)?120A 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 …121A 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, …122In 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 \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …123Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]124Let \(x\), \(y\) 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 \(|x| + |y| + |z|\).125A 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.126(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 …127Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]128Let \(H\) 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 …129At 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 …130Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.131A 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 \(\frac{1}{a}\); …132Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.133Let \(A\), \(B\), \(C\) 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 …134Let \(a\), \(b\) 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 …135A 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 …136A 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 \(y\) …137Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]138Let \(n\) 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 …139Let \(\varrho\) 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 \[ \text{(a) } A = \mathbb{N}, \qquad \text{(b) } A = \mathbb{T}, \qquad \text{(c) } A = \mathbb{Q}, \] …140Find 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.141Let \(ABC\) 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 …142Each 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? …143It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.144In 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, …145Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.146Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]147Let \(M\) 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 …148Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.149The 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 …150Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).151Find 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 \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …152The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).153Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.154Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.155Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).156Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.157An 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 …158For 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.159A 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?160Determine 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.161Consider 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 \(p(m)\)?162Two 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 …163For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]164In 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 …165The 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?166How 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 …167On 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 …168Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …169The 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 …
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