Republicko takmicenje (Republic round) 2004 · Lemma

1GeometryProofLet \(ABCD\) be a trapezoid with \(AB \parallel CD\), and let \(P\) be a point on the extension of the diagonal \(AC\) beyond \(C\), so that \(C\) lies between \(A\) and \(P\). Let \(X\) and \(Y\) be the …2GeometryProofIn an equilateral triangle \(ABC\) the side has length \(|AB| = 2\). Let \(M\) and \(N\) be interior points of the side \(AB\) with \(|MN| = 1\). Prove that \[ \angle MCN > 30^\circ . \]3Number theoryFree answerHow many triples \((a, b, c)\) of positive integers are there such that \(2a + 1\) is divisible by \(b\), \(2b + 1\) is divisible by \(c\), and \(2c + 1\) is divisible by \(a\)?4CombinatoricsProofA \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …5CombinatoricsFree answerHow many integers \(n\) with \(10 \le n < 100000\) are divisible by \(4\), contain no digit \(0\) in their decimal representation, and have no two adjacent digits equal?

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