Practice library
Problems
1Find every real number \(x\) that satisfies the equation \[ \bigl|\,2006 - |206 - x|\,\bigr| = 26 . \]Open2Prove 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?3Let \(t\) 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 \(t\).4Let \(x\) and \(y\) be real numbers satisfying \(y - x = 2\). Compute the value of the expression \[ 2x^3 + y^3 - 3y^2x + 6x^2 . \]5Find all pairs of real numbers \(x\) and \(y\) that solve the system \[ \frac{x}{6} + \frac{4}{y} = 2, \qquad \frac{18}{x} + \frac{y}{2} = 5 . \]6Let \(a\), \(b\), \(c\) be positive numbers with \(a^2 + c^2 = 2b^2\). Prove that \[ \frac{2}{a+c} = \frac{1}{a+b} + \frac{1}{b+c} . \]7Find every real number \(x\) that satisfies \[ \Bigl|\, \bigl|\, |2-x| - x \,\bigr| - 8 \,\Bigr| \le 2008 . \]8A 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 …9Find all solutions of the equation \[ x = \bigl|\,2x - |60 - 2x|\,\bigr| . \]10Jure 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 …11Find all real numbers \(x\) that satisfy the inequality \[ \frac{|x-3| + x}{x+1} < 1 . \]12Suppose the numbers \(a\) and \(b\) satisfy \(a + b = 1\) and \(ab \neq 0\). Prove that \[ \frac{a}{b^3 - 1} - \frac{b}{a^3 - 1} = \frac{2(b-a)}{a^2b^2 + 3} . \]13Andrej 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 …14Let \(a\) 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.15The real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).16For 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 \((x, y)\)?17Find all pairs of real numbers \(x\), \(y\) that satisfy the system \[ \begin{aligned} \frac{3}{x-4y} + \frac{2}{x+y-5} &= 0, \\ \frac{2}{x^{2}-4y^{2}} + \frac{1}{x^{2}+y^{2}-5} &= 0. \end{aligned} \]18The 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.
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