USAMO 1997

Let $ABC$ be a triangle. Take points $D$ , $E$ , $F$ on the perpendicular bisectors of $BC$ , $CA$ , $AB$ respectively. Show that the lines through $A$ , $B$ , $C$ perpendicular to $EF$ , $FD$ , $DE$ respectively are concurrent.

Prove that for any integer $n$ , there exists a unique polynomial $Q$ with coefficients in $\{0,1,\ldots,9\}$ such that $Q(-2) = Q(-5) = n$ .

To clip a convex $n$ -gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$ , and $NC$ , where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$ . In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$ -gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$ . Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$ , and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$ , for all $n \geq 6$ .

Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies $$a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1$$ for all $i,j \geq 1$ with $i + j \leq 1997$ . Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$ ) for all $1 \leq n \leq 1997$ .