USAMO 1976

The squares of a $4 \times 7$ chess board are colored red or blue. Show that however the coloring is done, we can find a rectangle with four distinct corner squares all the same color. Find a counter-example to show that this is not true for a $4 \times 6$ board.

$\mathrm{AB}$ is a fixed chord of a circle, not a diameter. $\mathrm{CD}$ is a variable diameter. Find the locus of the intersection of $\mathrm{AC}$ and $\mathrm{BD}$.

Find all integral solutions to $a^{2}+b^{2}+c^{2}=a^{2} b^{2}$.

A tetrahedron ABCD has edges of total length 1. The angles at A (BAC etc) are all $90^{\circ}$. Find the maximum volume of the tetrahedron.

The polynomials $a(x), b(x), c(x), d(x)$ satisfy $a\left(x^{5}\right)+x b\left(x^{5}\right)+x^{2} c\left(x^{5}\right)=\left(1+x+x^{2}+x^{3}+x^{4}\right) d(x)$. Show that $\mathrm{a}(\mathrm{x})$ has the factor $(\mathrm{x}-1)$.