2014 EGMO Problems
Contents
[hide]Day 1
Problem 1
Determine all real constants such that whenever
,
and
are the lengths of sides of a triangle, then so are
,
,
.
Problem 2
Let and
be points in the interiors of sides
and
, respectively, of a triangle
, such that
. Let the lines
and
meet at
. Prove that the incentre
of triangle
, the orthocentre
of triangle
and the midpoint
of the arc
of the circumcircle of triangle
are collinear.
Problem 3
We denote the number of positive divisors of a positive integer by
and the number of distinct prime divisors of
by
. Let
be a positive integer. Prove that there exist infinitely many positive integers
such that
and
does not divide
for any positive integers
satisfying
.
Day 2
Problem 4
Determine all positive integers for which there exist integers
satisfying the condition that if
and
divides
, then
.
Problem 5
Let be a positive integer. We have
boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
Problem 6
Determine all functions satisfying the condition
for all real numbers
and
.