USAMO 1975

Show that for any non-negative reals $\mathrm{x}, \mathrm{y},[5 \mathrm{x}]+[5 \mathrm{y}] \geq[3 \mathrm{x}+\mathrm{y}]+[\mathrm{x}+3 \mathrm{y}]$. Hence or otherwise show that (5a)! $(5 b)!/(a!b!(3 a+b)!(a+3 b)!)$ is integral for any positive integers $a, b$.

Show that for any tetrahedron the sum of the squares of the lengths of two opposite edges is at most the sum of the squares of the other four.

A polynomial $p(x)$ of degree $n$ satisfies $p(0)=0, p(1)=1 / 2, p(2)=2 / 3, \ldots, p(n)=n /(n+1)$. Find $p(n+1)$.

Two circles intersect at two points, one of them $\mathrm{X}$. Find $\mathrm{Y}$ on one circle and $\mathrm{Z}$ on the other, so that $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ are collinear and $\mathrm{XY} . \mathrm{XZ}$ is as large as possible.

A pack of $\mathrm{n}$ cards, including three aces, is well shuffled. Cards are turned over in turn. Show that the expected number of cards that must be turned over to reach the second ace is $(\mathrm{n}+1) / 2$.