Olympiad problems

2013 EGMO, 1

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

2013 EGMO, 6

Tags: pigeonhole principle , combinatorics , EGMO , EGMO 2013 , Hi

Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine. Prove that, on one of these $16$ days, all seven dwarves were collecting berries.

2013 EGMO, 2

Tags: geometry , rectangle , combinatorial geometry , EGMO , EGMO 2013

Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3,\ldots,10$ in some order.

2013 EGMO, 3

Tags: least common multiple , number theory , Combinatorial Number Theory , EGMO , EGMO 2013

Let $n$ be a positive integer. (a) Prove that there exists a set $S$ of $6n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32n^2$. (b) Prove that every set $T$ of $6n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9n^2$.