USAMO 1999

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions:

(a) every square that does not contain a checker shares a side with one that does;

(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.

Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

Let $ABCD$ be a cyclic quadrilateral. Prove that $$|AB - CD| + |AD - BC| \geq 2|AC - BD|.$$

Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$ , such that $$\left\{ \dfrac{ra}{p} \right\} + \left\{ \dfrac{rb}{p} \right\} + \left\{ \dfrac{rc}{p} \right\} + \left\{ \dfrac{rd}{p} \right\} = 2$$ for any integer $r$ not divisible by $p$ . Prove that at least two of the numbers $a+b$ , $a+c$ , $a+d$ , $b+c$ , $b+d$ , $c+d$ are divisible by $p$ . (Note: $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$ .)

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of $n>3$ real numbers such that $$a_{1}+\cdots+a_{n} \geq n \text { and } a_{1}^{2}+\cdots+a_{n}^{2} \geq n^{2}$$ Prove that $\max \left(a_{1}, \ldots, a_{n}\right) \geq 2$.

The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.

Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$ . The inscribed circle $\omega$ of triangle $BCD$ meets $CD$ at $E$ . Let $F$ be a point on the (internal) angle bisector of $\angle DAC$ such that $EF \perp CD$ . Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$ . Prove that the triangle $AFG$ is isosceles.