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Problems
1An 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 …Open2A 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 \(\tfrac{1}{8}\) …3What 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 \(100\).4A 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 …5Birds 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 …6Let \(a\) 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 \(\dfrac{a}{b} + \dfrac{b}{a} - \dfrac{1}{ab}\) …7The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]8Let \(a\) and \(b\) be real numbers for which \[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \] Prove that \[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]9Find all real numbers \(x\) for which \[ \Biggl|\, \Bigl|\, \bigl|\, |x| - 2 \,\bigr| - 20 \,\Bigr| - 200 \,\Biggr| = 2007 . \]10Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]11Real 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 \(\dfrac{b}{a}\). …12What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)13Let \(a\), \(b\), \(c\) 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 …14Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).15Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …16Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] 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 …17Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …18Tina 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 …19A 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 …20The 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 …21Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]22For 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 \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …23The 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 \(x - y\).24Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]25Find every real number \(x\) for which \[ \left(2x^{2} + 7x + 6\right)^{3} - \left(x^{2} + 3x + 2\right)^{3} = \left(x^{2} + 4x + 4\right)\left(117x^{2} + 128x + 52\right) . \]26How 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 \(x \in \mathbb{R}^{+}\)?27For 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 …28Positive 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 \(a = b = c = d = e\).29Decide 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 …30The 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?31Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \frac{3xy}{x-y} = 2, \qquad \frac{2yz}{y+2z} = 3, \qquad \frac{xz}{z-4x} = 3 . \]32Find the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …33Determine all polynomials \(R(x)\) whose coefficients all belong to the set \(\{-1, 1\}\) and which satisfy \[ R(3) = 130 \qquad \text{and} \qquad R(-2) = -45 . \]34Determine every natural number \(n\) for which there exist real numbers \(a\), \(b\), \(c\) satisfying \[ \bigl\{\, a + b + c,\; ab + bc + ca,\; abc \,\bigr\} = \{\, n,\; n+1,\; n+2 \,\} . \]35Let \(P(x)\) 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.36Let \(a\), \(b\), \(c\) 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 \(ab\), \(ac\), …37Find all pairs of real numbers \(x\) and \(y\) that satisfy the system \[ \frac{y^{3} + 15x^{2}}{y^{4} - x^{3}} = \frac{y^{2} + 15x}{y^{3} - x^{2}} , \] \[ \frac{1500y^{3} + 4x^{2}}{9y^{4} - 4} = \frac{1500y^{2} + 4x}{9y^{3} - 4} . \] …38Determine 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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