IMO 2014 - Competition day 1

The first day of the IMO has just drawn to a close, and here are the questions if you want to give them a shot!

Problem 1 (submitted by Austria)

Let \(a_0 < a_1 < a_2 \ldots\) be an infinite sequence of positive integers. Prove that there exists a unique integer \(n\geq 1\) such that \(a_n < \dfrac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}\).

Problem 2 (submitted by Croatia)

Let \(n \ge 2\) be an integer. Consider an \(n \times n\) chessboard consisting of \(n^2\) unit squares. A configuration of \(n\) rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer \(k\) such that, for each peaceful configuration of n rooks, there is a \(k \times k\) square which does not contain a rook on any of its \(k^2\) unit squares.

Problem 3 (submitted by Iran)

Convex quadrilateral \(ABCD\) has \(\angle ABC = \angle CDA = 90^{\circ}\). Point \(H\) is the foot of the perpendicular from \(A\) to \(BD\). Points \(S\) and \(T\) lie on sides \(AB\) and \(AD\), respectively, such that \(H\) lies inside triangle \(SCT\) and \(\angle CHS - \angle CSB = 90^{\circ}\), \(\quad \angle THC - \angle DTC = 90^{\circ}\). Prove that line \(BD\) is tangent to the circumcircle of triangle \(TSH\).