Compete · Slovenia
Regijsko tekmovanje (regional round) 2013
1A 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 …2The 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 \(1\)3The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …4Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]5Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …6Benjamin was working out the sum \(1 + 2 + 3 + \dots + 2012\). He left out a few of the terms, and the wrong total he ended up with was divisible by \(2011\). Anika was working out the sum \[ A = 1 + 2 + 3 + \dots + 2013 , \] …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.