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1Let \(a\), \(b\), \(c\), \(d\) 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 \[ I = a^{2} + b^{2} + c^{2} + d^{2} + e^{2}. \] …Open2Determine all functions \(f : \mathbb{R} \to \mathbb{R}\) such that for every real number \(x\), \[ f(x+1) \le x \le f(x) + 1. \]3The 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 …4Solve 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 …5Find the remainder on dividing the polynomial \(x^{2011} + 1\) by the polynomial \((x+1)^2\).6Find all real numbers \(x\) for which \[ \bigl|\,|x| - 1\,\bigr| + \bigl|\,|x| + 2\,\bigr| = 3. \]7Several 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?8Real 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.9If \(a^{2} + a + 1 = 0\), what is the value of \[ a^{1995} + \frac{1}{a^{1995}} \, ? \]10Determine 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.11Real 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 …12Let \(f\) 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 …13A polynomial \(P\) with integer coefficients satisfies \[ P\bigl(P(2023) + 2023\bigr) = 1. \] Which values can the number \(P(2023)\) take?14Find 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.15It 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) …16Does 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 \(x\)?17Determine 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 \(1, 2, \dots, 2017\) …18Find 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 …19Let \(a\), \(b\) 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 \(|p - q| < 1\).20Find all pairs of real numbers \((x, y)\) satisfying \[ \frac{|x+y|}{1+|x+y|} = \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}. \]21Find all values \(a \in \mathbb{R}\) for which the equation \[ |x - a| + |a - 1| = 1 \] has two solutions, and determine those solutions.22The 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 \[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \] …
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