2013 EGMO Problems
Contents
[hide]Day 1
Problem 1
The side 
of the triangle 
is extended beyond 
to 
so that 
. The side 
is extended beyond 
to 
so that 
. Prove that, if 
, then the triangle is right-angled.
Problem 2
Determine all integers 
for which the 
square can be dissected into five rectangles, the side lengths of which are the integers 
in some order.
Problem 3
Let 
be a positive integer.
(a) Prove that there exists a set 
of 
pairwise different positive integers, such that the least common multiple of any two elements of is no larger than 
.
(b) Prove that every set 
of pairwise different positive integers contains two elements the least common multiple of which is larger than 
.
Day 2
Problem 4
Find all positive integers 
and 
for which there are three consecutive integers at which the polynomial![\[P(n) = \frac{n^5+a}{b}\]](http://latex.artofproblemsolving.com/b/9/9/b99aee376ddf19d29c9648e9561fe961e20ae0bf.png)
takes integer values.
Problem 5
Let 
be the circumcircle of the triangle . The circle 
is tangent to the sides 
and , and it is internally tangent to the circle at the point 
. A line parallel to 
intersecting the interior of triangle is tangent to at 
.
Prove that 
.
Problem 6
Snow White and the Seven Dwarves are living in their house in the forest. On each of 
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 days, all seven dwarves were collecting berries.
