2012 EGMO Problems
Contents
[hide]Day 1
Problem 1
Let be a triangle with circumcentre . The points lie in the interiors of the sides respectively, such that is perpendicular to and is perpendicular to . (By interior we mean, for example, that the point lies on the line and is between and on that line.) Let be the circumcentre of triangle . Prove that the lines and are perpendicular.
Problem 2
Let be a positive integer. Find the greatest possible integer , in terms of , with the following property: a table with rows and columns can be filled with real numbers in such a manner that for any two different rows and the following holds: Solution
Problem 3
Find all functions such thatfor all .
Problem 4
A set of integers is called sum-full if , i.e. each element is the sum of some pair of (not necessarily different) elements . A set of integers is said to be zero-sum-free if is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of . Does there exist a sum-full zero-sum-free set of integers?
Day 2
Problem 5
The numbers and are prime and satisfy for some positive integer . Find all possible values of .
Problem 6
There are infinitely many people registered on the social network Mugbook. Some pairs of (different) users are registered as friends, but each person has only finitely many friends. Every user has at least one friend. (Friendship is symmetric; that is, if is a friend of , then is a friend of .) Each person is required to designate one of their friends as their best friend. If designates as her best friend, then (unfortunately) it does not follow that necessarily designates as her best friend. Someone designated as a best friend is called a -best friend. More generally, if is a positive integer, then a user is an -best friend provided that they have been designated the best friend of someone who is an -best friend. Someone who is a -best friend for every positive integer is called popular.
(a) Prove that every popular person is the best friend of a popular person.
(b) Show that if people can have infinitely many friends, then it is possible that a popular person is not the best friend of a popular person.
Problem 7
Let be an acute-angled triangle with circumcircle and orthocentre . Let be a point of on the other side of from . Let be the reflection of in the line , and let be the reflection of in the line . Let be the second point of intersection of with the circumcircle of triangle . Show that the lines , and are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)
Problem 8
A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example, and are repetitive, but and are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)