2004 USAMO Problems

Problems from the 2004 USAMO.

Day 1

Problem 1

Let $\displaystyle ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least $\displaystyle 60 ^{\circ}$ . Prove that

$\frac{1}{3} | AB^3 - AD^3 | \le | BC^3 - CD^3 | \le 3 |AB^3 - AD^3 |$

When does equality hold?

Solution

Problem 2

Suppose $a_1, \ldots, a_n$ are integers whose greatest common divisor is 1. Let $\displaystyle S$ be a sequence of integers with the following properties:

(a) For $i = 1, \ldots , n$ , $a_i \in S$ .
(b) For $i, j = 1, \ldots, n$ (not necessarily distinct), $a_i - a_j \in S$ .
(c) For any integers $x, y \in S$ , if $x+y \in S$ , then $x-y \in S$ .

Prove that $\displaystyle S$ must be the set of all integers.

Solution

Problem 3

For what values of $\displaystyle k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons?

Solution

Day 2

Problem 4

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares. Find, with proof, a winning strategy for one of the players.

Solution

Problem 5

Let $\displaystyle a$ , $\displaystyle b$ , and $\displaystyle c$ be positive real numbers. Prove that

$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3$ .

Problem 6

A circle $\displaystyle \omega$ is inscribed in a quadrilateral $\displaystyle ABCD$ . Let $\displaystyle I$ be the center of $\displaystyle \omega$ . Suppose that