USAMO 1980

A balance has unequal arms and pans of unequal weight. It is used to weigh two objects of unequal weight. The first object balances against a weight A, when placed in the left pan and against a weight a, when placed in the right pan. The corresponding weights for the second object are $\mathrm{B}$ and $\mathrm{b}$. A third object balances against a weight C, when placed in the left pan. What is its true weight?

Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals.

A $+B+C$ is an integral multiple of $\pi . x, y, z$ are real numbers. If $x \sin A+y \sin B+z \sin C=x^{2} \sin 2 A+$ $y^{2} \sin 2 B+z^{2} \sin 2 C=0$, show that $x^{n} \sin n A+y^{n} \sin n B+z^{n} \sin n C=0$ for any positive integer $n$.

The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.

If $x, y, z$ are reals such that $0 \leq x, y, z \leq 1$, show that $x /(y+z+1)+y /(z+x+1)+z /(x+y+1) \leq 1-(1-$ $\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})$.