Practice library
Problems
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 every real number \(x\) that satisfies the equation \[ \bigl|\,2006 - |206 - x|\,\bigr| = 26 . \]22Prove 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?23Let \(t\) 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 \(t\).24Find all values \(a \in \mathbb{R}\) for which the equation \[ |x - a| + |a - 1| = 1 \] has two solutions, and determine those solutions.25The 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 \[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \] …26Let \(x\) and \(y\) be real numbers satisfying \(y - x = 2\). Compute the value of the expression \[ 2x^3 + y^3 - 3y^2x + 6x^2 . \]27Find all pairs of real numbers \(x\) and \(y\) that solve the system \[ \frac{x}{6} + \frac{4}{y} = 2, \qquad \frac{18}{x} + \frac{y}{2} = 5 . \]28Let \(a\), \(b\), \(c\) be positive numbers with \(a^2 + c^2 = 2b^2\). Prove that \[ \frac{2}{a+c} = \frac{1}{a+b} + \frac{1}{b+c} . \]29Find every real number \(x\) that satisfies \[ \Bigl|\, \bigl|\, |2-x| - x \,\bigr| - 8 \,\Bigr| \le 2008 . \]30A 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 …31Find all solutions of the equation \[ x = \bigl|\,2x - |60 - 2x|\,\bigr| . \]32Find 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 \[ Q(x) = x^{2} - (a+1)x + a. \]33Jure 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 …34Find all real numbers \(x\) that satisfy the inequality \[ \frac{|x-3| + x}{x+1} < 1 . \]35The 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 \(2012^{12}\) …36A 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 …37Let \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]38Let \(p\), \(q\), \(r\) 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 \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …39Aca, 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 …40Suppose the numbers \(a\) and \(b\) satisfy \(a + b = 1\) and \(ab \neq 0\). Prove that \[ \frac{a}{b^3 - 1} - \frac{b}{a^3 - 1} = \frac{2(b-a)}{a^2b^2 + 3} . \]41Andrej 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 …42Let \(a\) 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.43The 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 \(1\)44The real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).45Nonzero 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 \(a = 3c\) …46Peter 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 …47Let \(x\), \(y\) 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 \(|x| + |y| + |z|\).48A 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 \(\frac{1}{a}\); …49A 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 \(y\) …50Find 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.51Each 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? …52For 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 \((x, y)\)?53Find all pairs of real numbers \(x\), \(y\) that satisfy the system \[ \begin{aligned} \frac{3}{x-4y} + \frac{2}{x+y-5} &= 0, \\ \frac{2}{x^{2}-4y^{2}} + \frac{1}{x^{2}+y^{2}-5} &= 0. \end{aligned} \]54During 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 …55The 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 \(60\) …56Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).57The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).58Determine 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.59How 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 …60Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …61The 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.62Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]63Consider 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 …64Let \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …65Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …66Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]67Two 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 …68Let \(x, y \in \mathbb{R}\) 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, …69Ana 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 …70Determine 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 \((x, y, z)\) …71Which of the following two numbers is greater: \[ \frac{1.\underbrace{11\ldots1}_{2005 \text{ digits}}}{1.\underbrace{11\ldots1}_{2006 \text{ digits}}} \qquad \text{or} \qquad \frac{1.\underbrace{0101\ldots01}_{4010 \text{ digits}}}{1.\underbrace{0101\ldots01}_{4012 \text{ digits}}} \, ? \] …72An 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 …73A 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 \(\tfrac{1}{8}\) …74What 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 \(100\).75A 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 …76Let \(a\), \(b\), \(c\), \(d\) 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 \(\dfrac{9a+b}{9c+d}\). …77Let \(a\), \(b\), \(c\) be positive numbers with \(a > c\) and \(b > c\). Prove that \[ \sqrt{c(a-c)} + \sqrt{c(b-c)} \;\leq\; \sqrt{ab} . \]78Birds 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 …79Find 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 …80Let \(a\) 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 \(\dfrac{a}{b} + \dfrac{b}{a} - \dfrac{1}{ab}\) …81Let \(a\) 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 \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …82Let \(a_1, a_2, \ldots, a_n\) 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 …83A 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 \(p(x)\).84The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]85Let \(a\) and \(b\) be real numbers for which \[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \] Prove that \[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]86Find all real numbers \(x\) for which \[ \Biggl|\, \Bigl|\, \bigl|\, |x| - 2 \,\bigr| - 20 \,\Bigr| - 200 \,\Biggr| = 2007 . \]87Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]88Real 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 \(\dfrac{b}{a}\). …89What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)90Ali-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, …91Real numbers \(a, b, c, d\) satisfy \[ a^{2} + b^{2} = c^{2} + d^{2} , \qquad ab + cd > 0 , \qquad ac + bd > 0 . \] Prove that \(ad + bc > 0\).92Let \(a\), \(b\), \(c\) 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 …93Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).94Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …95Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] 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 …96Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …97Let \(a\), \(b\), \(c\) 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 …98Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …99Let \(x\) 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?100Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]101Tina 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 …102A 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 …103The 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 …104Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]105For 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 \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …106The 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 \(x - y\).107Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]108Find every real number \(x\) for which \[ \left(2x^{2} + 7x + 6\right)^{3} - \left(x^{2} + 3x + 2\right)^{3} = \left(x^{2} + 4x + 4\right)\left(117x^{2} + 128x + 52\right) . \]109How 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 \(x \in \mathbb{R}^{+}\)?110For 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 …111Positive 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 \(a = b = c = d = e\).112Decide 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 …113The 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?114Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \frac{3xy}{x-y} = 2, \qquad \frac{2yz}{y+2z} = 3, \qquad \frac{xz}{z-4x} = 3 . \]115Find the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …116Determine all polynomials \(R(x)\) whose coefficients all belong to the set \(\{-1, 1\}\) and which satisfy \[ R(3) = 130 \qquad \text{and} \qquad R(-2) = -45 . \]117Determine every natural number \(n\) for which there exist real numbers \(a\), \(b\), \(c\) satisfying \[ \bigl\{\, a + b + c,\; ab + bc + ca,\; abc \,\bigr\} = \{\, n,\; n+1,\; n+2 \,\} . \]118Let \(P(x)\) 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.119Let \(a\), \(b\), \(c\) 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 \(ab\), \(ac\), …120Find all pairs of real numbers \(x\) and \(y\) that satisfy the system \[ \frac{y^{3} + 15x^{2}}{y^{4} - x^{3}} = \frac{y^{2} + 15x}{y^{3} - x^{2}} , \] \[ \frac{1500y^{3} + 4x^{2}}{9y^{4} - 4} = \frac{1500y^{2} + 4x}{9y^{3} - 4} . \] …121Determine 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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