2013 USAJMO

Day 1

Problem 1

Are there integers and such that and are both perfect cubes of integers?

Solution

Problem 2

Each cell of an board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:

(i) The difference between any two adjacent numbers is either or . (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to .

Determine the number of distinct gardens in terms of and .

Solution

Problem 3

In triangle , points lie on sides respectively. Let , , denote the circumcircles of triangles , , , respectively. Given the fact that segment intersects , , again at respectively, prove that .

Solution

Day 2

Problem 4

Let be the number of ways to write as a sum of powers of , where we keep track of the order of the summation. For example, because can be written as , , , , , and . Find the smallest greater than for which is odd.

Solution

Problem 5

Quadrilateral is inscribed in the semicircle with diameter . Segments and meet at . Point is the foot of the perpendicular from to line . Point lies on such that line is perpendicular to line . Let be the intersection of segments and . Prove that

Solution

Problem 6

Find all real numbers satisfying

Solution

See Also

2013 USAJMO (Problems • Resources)
Preceded by 2012 USAJMO	Followed by 2014 USAJMO
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions