Compete · Serbia
Drzavno takmicenje (National round) 2013
1Let \(k > 0\). On the sides \(A_1B_1\), \(B_1C_1\) and \(C_1A_1\) of a triangle \(A_1B_1C_1\), points \(C_2\), \(A_2\) and \(B_2\) are chosen, respectively, so that \[ \frac{A_1C_2}{C_2B_1} = \frac{B_1A_2}{A_2C_1} = \frac{C_1B_2}{B_2A_1} = k . \] …2Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).3Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(abcd = 1\) and \[ a + b + c + d = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} . \] Prove that some two of the numbers \(ab\), \(ac\), …4On \(41\) squares of a chessboard - the ordinary \(8 \times 8\) board - a king is placed, one king on each of those squares. Prove that among these kings one can find three pairwise disjoint sets, each …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.