USAMO 1999 — Days 1 and 2

Some checkers on an $n\times n$ board satisfy:
(a) every empty square shares a side with an occupied square;
(b) between any two occupied squares there is a sequence of occupied squares joining them, with consecutive squares sharing a side.
Prove that at least $(n^2-2)/3$ checkers have been placed.

Let $ABCD$ be a convex cyclic quadrilateral. Prove
\[|AB-CD|+|AD-BC|\ge2|AC-BD|.\]

Let $p>2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$. Suppose that for every integer $r$ not divisible by $p$,
\[\left\{\frac{ra}{p}\right\}+\left\{\frac{rb}{p}\right\}+\left\{\frac{rc}{p}\right\}+\left\{\frac{rd}{p}\right\}=2,\]
where $\{t\}=t-\lfloor t\rfloor$ is the fractional part. Prove that at least two of $a+b,a+c,a+d,b+c,b+d,c+d$ are divisible by $p$.

Let $a_1,\ldots,a_n$ be $n>3$ real numbers such that $\sum_{i=1}^na_i\ge n$ and $\sum_{i=1}^na_i^2\ge n^2$. Prove that $\max(a_1,\ldots,a_n)\ge2$.

The Y2K Game is played on a $1\times2000$ row. Two players alternate writing either S or O in an empty square. The first player to produce three consecutive squares spelling SOS wins. If the board fills without SOS, 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 incircle $\omega$ of triangle $BCD$ touches $CD$ at $E$. Let $F$ lie on the internal angle bisector of $\angle DAC$ with $EF\perp CD$. The circumcircle of triangle $ACF$ meets line $CD$ at $C$ and $G$. Prove that triangle $AFG$ is isosceles.