Republicko takmicenje (Republic round) 2006 · Lemma

1CombinatoricsProofThe points of a plane \(\alpha\) are split between two nonempty sets \(A\) and \(B\): no point belongs to both, and every point belongs to one of them. Prove that some isosceles right triangle has all …2Number theoryProofFind every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]3GeometryProofA pentagon \(ABCDE\) is inscribed in a circle of radius \(r\), and three of its sides have length \(r\): \[ AB = BC = DE = r . \] Let \(G\) and \(F\) be the midpoints of the sides \(CD\) and \(EA\). Prove …4GeometryProofProve that among all triangles with one and the same perimeter, the equilateral triangle has the largest area.5Number theoryFree answerFor some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?

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