# 2003 USAMO Problems

## Contents

## Day 1

### Problem 1

Prove that for every positive integer there exists an -digit number divisible by all of whose digits are odd.

### Problem 2

A convex polygon in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.

### Problem 3

Let . For every sequence of integers

satisfying , for , define another sequence

by setting to be the number of terms in the sequence that precede the term and are different from . Show that, starting from any sequence as above, fewer than applications of the transformation lead to a sequence such that .

## Day 2

### Problem 4

Let be a triangle. A circle passing through and intersects segments and at and , respectively. Lines and intersect at , while lines and intersect at . Prove that if and only if .

### Problem 5

Let , , be positive real numbers. Prove that

### Problem 6

At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

## Resources

- USAMO Problems and Solutions
- 2003 USAMO Problems and Solutions
- 2003 USAMO Problems on the Resources page

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.