Compete · Serbia
Opštinsko takmičenje (Municipal round) 2008
1Determine whether the number \[ 10^{5^{10^{5^{10}}}} + 5^{10^{5^{10^{5}}}} \] is divisible by \(11\).2Determine 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.3One afternoon Ana and Olja each walked in a straight line to visit her boyfriend: Ana to Kosta's house, Olja to Laza's house. The two routes crossed at an old tree, and there the girls met. Standing under …4It 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) …5There are \(14\) books standing in a row on a shelf. In how many ways can \(5\) of them be chosen so that no two of the chosen books stand next to each other?
Work through the paper in order - each problem opens in the workspace with this paper as its trail.