# 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 follows 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.