USAMO 1973

Show that if two points lie inside a regular tetrahedron the angle they subtend at a vertex is less than $\pi / 3$.

The sequence $a_{n}$ is defined by $a_{1}=a_{2}=1, a_{n+2}=a_{n+1}+2 a_{n}$. The sequence $b_{n}$ is defined by $b_{1}=1, b_{2}=7$, $b_{n+2}=2 b_{n+1}+3 b_{n}$. Show that the only integer in both sequences is 1 .

Three vertices of a regular $2 n+1$ sided polygon are chosen at random. Find the probability that the center of the polygon lies inside the resulting triangle.

Find all complex numbers $x, y, z$ which satisfy $x+y+z=x^{2}+y^{2}+z^{2}=x^{3}+y^{3}+z^{3}=3$.

Show that the cube roots of three distinct primes cannot be terms in an arithmetic progression (whether consecutive or not).