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 polynomialtakes 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.