USAMO 1996 — Days 1 and 2

Prove that the average of the numbers $n\sin n^\circ$ for $n=2,4,6,\ldots,180$ is $\cot1^\circ$.

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of its elements. Given a set $A$ of $n$ positive integers, consider all distinct sums $\sigma(S)$ over nonempty subsets $S\subseteq A$. Prove that these sums can be partitioned into $n$ classes so that in each class the ratio of the largest sum to the smallest sum is at most $2$.

Let $ABC$ be a triangle. Prove that there is a line $\ell$ in its plane such that the intersection of the interior of $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area greater than $\frac23$ the area of $ABC$.

A binary sequence of length $n$ is a sequence $(x_1,\ldots,x_n)$ whose terms are $0$ or $1$. Let $a_n$ count such sequences with no consecutive pattern $010$. Let $b_n$ count such sequences with neither consecutive pattern $0011$ nor consecutive pattern $1100$. Prove that $b_{n+1}=2a_n$ for every positive integer $n$.

Let $ABC$ be a triangle and let $M$ be an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$, and $\angle MCA=30^\circ$. Prove that $ABC$ is isosceles.

Determine, with proof, whether there is a subset $X\subseteq\mathbb Z$ such that for every $n\in\mathbb Z$, the equation $a+2b=n$ has exactly one solution with $a,b\in X$.