# 2010-2011 Mock USAJMO Problems/Solutions

## Contents

## Problems

### Problem 1

Given two fixed, distinct points and on plane , find the locus of all points belonging to such that the quadrilateral formed by point , the midpoint of , the centroid of , and the midpoint of (in that order) can be inscribed in a circle.

### Problem 2

Let be positive real numbers such that . Prove that with equality if and only if .

### Problem 3

At a certain (lame) conference with people, each group of three people wishes to meet exactly once. On each of the days of the conference, every person participates in at most one meeting with two other people, with no limit on the number of groups that can meet. Determine for

a)

b)

### Problem 4

The sequence satisfies and for . Prove that for all nonnegative integers , we have

### Problem 5

Find all solutions to where and are positive integers.

### Problem 6

In convex, tangential quadrilateral , let the incircle touch the four sides at points , respectively. Prove that
**Note:** A tangential quadrilateral has concurrent angle bisectors, and denotes the area of figure .