Okruzno takmicenje (Regional round) 2003 · Lemma

1Number theoryProofLet \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).2Number theoryFree answerFind all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]3Number theoryProofProve that for every integer \(n \geqslant 2\) one can find \(n\) pairwise distinct positive integers whose squares add up to the square of a positive integer.4GeometryProofLet \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that \[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \] and …5GeometryProofTwo points \(A_1\), \(B_1\) and a line \(p\) are given in the plane. Construct a triangle \(ABC\) in which \(A_1\) is the midpoint of the side \(BC\), \(B_1\) is the midpoint of the side \(CA\), and the …

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