Drzavno tekmovanje (national round) 2001 · Lemma

1Number theoryProofLet \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers \(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\) is divisible by \(5\). …2AlgebraFree answerTina 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 …3GeometryProofLet \(AB\) be a segment and let \(P\) be any point of it other than \(A\) and \(B\). On the hypotenuses \(AP\) and \(PB\) erect isosceles right triangles \(APQ\) and \(PBR\), with the right angles at the …4CombinatoricsFree answerAndraz and Breda cut two long strips out of a newspaper, of lengths \(a\) and \(b\), to play a game with. A move consists of choosing one of the strips and cutting a piece of length \(d\) off it, so that …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.