USAMO 1974

$p(x)$ is a polynomial with integral coefficients. Show that there are no solutions to the equations $p(a)=b$, $\mathrm{p}(\mathrm{b})=\mathrm{c}, \mathrm{p}(\mathrm{c})=\mathrm{a}$, with $\mathrm{a}, \mathrm{b}, \mathrm{c}$ distinct integers.

Show that for any positive reals $\mathrm{x}, \mathrm{y}, \mathrm{z}$ we have $\mathrm{x}^{\mathrm{x}} \mathrm{y}^{\mathrm{y}} \mathrm{z}^{\mathrm{z}} \geq(\mathrm{xyz})^{\mathrm{a}}$, where $\mathrm{a}$ is the arithmetic mean of $\mathrm{x}, \mathrm{y}, \mathrm{z}$.

Two points in a thin spherical shell are joined by a curve shorter than the diameter of the shell. Show that the curve lies entirely in one hemisphere.

A, B, C play a series of games. Each game is between two players, The next game is between the winner and the person who was not playing. The series continues until one player has won two games. He wins the series. A is the weakest player, C the strongest. Each player has a fixed probability of winning against a given opponent. A chooses who plays the first game. Show that he should choose to play himself against B.

A point inside an equilateral triangle with side 1 is a distance $\mathrm{a}, \mathrm{b}, \mathrm{c}$ from the vertices. The triangle $\mathrm{ABC}$ has $\mathrm{BC}=\mathrm{a}, \mathrm{CA}=\mathrm{b}, \mathrm{AB}=\mathrm{c}$. The sides subtend equal angles at a point inside it. Show that sum of the distances of the point from the vertices is 1 .