2015 USAMO Problems
Contents
[hide]Day 1
Problem 1
Solve in integers the equation
Problem 2
Quadrilateral is inscribed in circle
with
and
. Let
be a variable point on segment
. Line
meets
again at
(other than
). Point
lies on arc
of
such that
is perpendicular to
. Let
denote the midpoint of chord
. As
varies on segment
, show that
moves along a circle.
Problem 3
Day 2
Problem 4
Steve is piling indistinguishable stones on the squares of an
grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions
for some
, such that
and
. A stone move consists of either removing one stone from each of
and
and moving them to
and
respectively,j or removing one stone from each of
and
and moving them to
and
respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?