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1Ten 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 …Open2Two circles that do not intersect are given. Construct all of their common tangent lines.3Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]4Find 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 \[ Q(x) = x^{2} - (a+1)x + a. \]5In 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 …6A 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, …7On 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) …8Prove 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.9Find 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 \(1\).10Let \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]11Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).12The 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 …13Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).14The 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 …15Let \(p\), \(q\), \(r\) 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 \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …16Let \(x\), \(y\) 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 \(x + y + z\).17Aca, 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 …18Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]19Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).20Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …21How 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 \(3\)?22A 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 …23A 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, …24In 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 \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …25Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]26Let \(x\), \(y\) 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 \(|x| + |y| + |z|\).27A 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.28(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 …29Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]30Let \(H\) 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 …31At 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 …32Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.33A 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 \(\frac{1}{a}\); …34Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.35Let \(A\), \(B\), \(C\) 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 …36Let \(a\), \(b\) 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 …37A 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 …38A 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 \(y\) …39Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]40Let \(n\) 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 …41Let \(\varrho\) 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 \[ \text{(a) } A = \mathbb{N}, \qquad \text{(b) } A = \mathbb{T}, \qquad \text{(c) } A = \mathbb{Q}, \] …42Find 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.43Let \(ABC\) 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 …44Each 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? …45It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.46In 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, …47Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.48Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]49Let \(M\) 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 …50Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.51The 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 …52Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).53Find 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 \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …54The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).55Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.56Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.57Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).58Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.59An 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 …60For 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.61A 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?62Determine 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.63Consider 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 \(p(m)\)?64Two 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 …65For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]66In 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 …67The 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?68How 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 …69On 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 …70Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …71The 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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