USAMO 2004 — Days 1 and 2

Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least $60$ degrees. Prove that
$$\frac13|AB^3-AD^3|\le|BC^3-CD^3|\le3|AB^3-AD^3|.$$
When does equality hold?

Proposed by Titu Andreescu (credited in the supplied source).

Let $a_1,a_2,\ldots,a_n$ be integers whose greatest common divisor is $1$. Let $S$ be a set of integers with the following properties:

(a) $a_i\in S$ for $i=1,\ldots,n$.

(b) $a_i-a_j\in S$ for $i,j=1,\ldots,n$, not necessarily distinct.

(c) If $x,y\in S$ and $x+y\in S$, then $x-y\in S$ too.

Prove that $S=\mathbb Z$.

Proposed by Kiran Kedlaya (credited in the supplied source).

For what real values of $k\gt 0$ is it possible to dissect a $1\times k$ rectangle into two similar but noncongruent polygons?

Proposed by Ricky Liu (credited in the supplied source).

Alice and Bob play a game on a $6$ by $6$ grid. On each turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square. Alice goes first and the players alternate. When the grid is full, the square with the greatest number in each row is colored black. Alice wins if a line can be drawn from the top of the grid to the bottom that stays in black squares; Bob wins otherwise. If two squares share a vertex, the line may pass from one to the other while staying in those two squares. Find, with proof, a winning strategy for one player.

Proposed by Melanie Wood (credited in the supplied source).

Let $a,b,c$ be positive real numbers. Prove that
$$(a^5-a^2+3)(b^5-b^2+3)(c^5-c^2+3)\ge(a+b+c)^3.$$

Proposed by Titu Andreescu (credited in the supplied source).

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be its center. Suppose that
$$(AI+DI)^2+(BI+CI)^2=(AB+CD)^2.$$
Prove that $ABCD$ is an isosceles trapezoid.

Proposed by Zuming Feng (credited in the supplied source).