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1The 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 \(2012^{12}\) …Open2A 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 …3The 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 \(1\)4Nonzero 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 \(a = 3c\) …5Peter 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 …6During 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 …7The 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 \(60\) …8Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]9Ana 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 …10Let \(a\) 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 \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …
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