# 1999 USAMO Problems

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Problems of the 1999 USAMO.

## Day 1

### Problem 1

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.

### Problem 2

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

### Problem 3

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 $x$.)

## Day 2

### Problem 4

Let $a_{1}, a_{2}, \dots, a_{n}$ ( $n > 3$) be real numbers such that $$a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}.$$ Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.

### Problem 5

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.

### Problem 6

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.

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