CMO 1992

Prove that the product of the first $n$ natural numbers is divisible by the sum of the first $n$ natural numbers if and only if $n+1$ is not an odd prime.

For $x,y,z \geq 0,$ establish the inequality

$$x(x-z)^2 + y(y-z)^2 \geq (x-z)(y-z)(x+y-z)$$

and determine when equality holds.

In the diagram, $ABCD$ is a square, with $U$ and $V$ interior points of the sides $AB$ and $CD$ respectively. Determine all the possible ways of selecting $U$ and $V$ so as to maximize the area of the quadrilateral $PUQV$ .

Solve the equation

$$x^2 + \frac{x^2}{(x+1)^2} = 3$$

A deck of $2n+1$ cards consists of a joker and, for each number between 1 and $n$ inclusive, two cards marked with that number. The $2n+1$ cards are placed in a row, with the joker in the middle. For each $k$ with $1 \leq k \leq n,$ the two cards numbered $k$ have exactly $k-1$ cards between them. Determine all the values of $n$ not exceeding 10 for which this arrangement is possible. For which values of $n$ is it impossible?