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
Let , where . Each of the subsets of is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set , we then write for the number of subsets of T that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets and of ,
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?