2015 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be an acute-angled triangle, and let be the foot of the altitude from The angle bisector of intersects at and meets the circumcircle of triangle again at If , show that is tangent to
Problem 2
A domino is a or tile. Determine in how many ways exactly dominoes can be placed without overlapping on a chessboard so that every square contains at least two uncovered unit squares which lie in the same row or column.
Problem 3
Let be integers greater than , and let be positive integers not greater than . Prove that there exist positive integers not greater than , such thatwhere denotes the greatest common divisor of .
Day 2
Problem 4
Determine whether there exists an infinite sequence of positive integers which satisfies the equalityfor every positive integer .
Problem 5
Let be positive integers with . Anastasia partitions the integers into pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to .
Problem 6
Let be the orthocentre and be the centroid of acute-angled triangle with . The line intersects the circumcircle of at and . Let be the reflection of in the line . Prove that if and only if