2011 USAJMO Problems
Contents
Day 1
Problem 1
Find, with proof, all positive integers for which is a perfect square.
Problem 2
Let , , be positive real numbers such that . Prove that
Problem 3
For a point in the coordinate plane, let denote the line passing through with slope . Consider the set of triangles with vertices of the form , , , such that the intersections of the lines , , form an equilateral triangle . Find the locus of the center of as ranges over all such triangles.
Day 2
Problem 4
A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words , , , be defined as follows: , , and for , is the word formed by writing followed by . Prove that for any , the word formed by writing , , , in succession is a palindrome.
Problem 5
Points , , , , lie on a circle and point lies outside the circle. The given points are such that (i) lines and are tangent to , (ii) , , are collinear, and (iii) . Prove that bisects .
Problem 6
Consider the assertion that for each positive integer , the remainder upon dividing by is a power of 4. Either prove the assertion or find (with proof) a counterexample.
See Also
2011 USAJMO (Problems • Resources) | ||
Preceded by 2010 USAJMO Problems |
Followed by 2012 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.