Okruzno takmicenje (Regional round) 1995 · Lemma

1Number theoryProofProve that a positive integer whose decimal representation uses no digits other than \(2\) and \(6\) cannot be written as a difference of the squares of two integers.2CombinatoricsProofAn infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]3AlgebraFree answerEach of the numbers \(1, 2, \dots, 1995\) is to be given a sign \(+\) or \(-\). How should the signs be chosen so that the value of \[ \pm 1 \pm 2 \pm \cdots \pm 1995 \] is as close to zero as possible? …4GeometryProofEach diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.5GeometryProofOn the sides of a triangle \(ABC\), equilateral triangles \(ADB\), \(BEC\) and \(CFA\) are constructed outwardly, so that \(D\), \(E\), \(F\) are the apexes over \(AB\), \(BC\), \(CA\) respectively. Prove …

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