# 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!

http://www.artofproblemsolving.com/Forum/viewforum.php?f=1098

## 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 $$TSH$$.