USAMO 1986

Do there exist 14 consecutive positive integers each divisible by a prime less than 13 ? What about 21 consecutive positive integers each divisible by a prime less than 17 ?

Five professors attended a lecture. Each fell asleep just twice. For each pair there was a moment when both were asleep. Show that there was a moment when three of them were asleep.

What is the smallest $\mathrm{n}>1$ for which the average of the first $\mathrm{n}$ (non-zero) squares is a square?

A T-square allows you to construct a straight line through two points and a line perpendicular to a given line through a given point. Circles $\mathrm{C}$ and $\mathrm{C}^{\prime}$ intersect at $\mathrm{X}$ and $\mathrm{Y}$. $\mathrm{XY}$ is a diameter of $\mathrm{C}$. $\mathrm{P}$ is a point on $\mathrm{C}^{\prime}$ inside

C. Using only a $T$-square, find points $\mathrm{Q}, \mathrm{R}$ on $\mathrm{C}$ such that $\mathrm{QR}$ is perpendicular to $\mathrm{XY}$ and $\mathrm{PQ}$ is perpendicular to PR.

A partition of $n$ is an increasing sequence of integers with sum $n$. For example, the partitions of 5 are: $1,1,1$, 1,$1 ; 1,1,1,2 ; 1,1,3 ; 1,4 ; 5 ; 1,2,2 ;$ and 2,3 . If $\mathrm{p}$ is a partition, $\mathrm{f}(\mathrm{p})=$ the number of $1 \mathrm{~s}$ in $\mathrm{p}$, and $\mathrm{g}(\mathrm{p})=$ the number of distinct integers in the partition. Show that $\sum f(p)=\sum g(p)$, where the sum is taken over all partitions of $n$.