# Canadian Mathematical Olympiad 

1992

## PROBLEM 1

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.

Problem 2
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.

## Problem 3

In the diagram, $A B C D$ is a square, with $U$ and $V$ interior points of the sides $A B$ and $C D$ respectively. Determine all the possible ways of selecting $U$ and $V$ so as to maximize the area of the quadrilateral $P U Q V$.
![](https://cdn.mathpix.com/cropped/2024_11_22_56f7ec16d00bdf6bd08eg-1.jpg?height=326&width=345&top_left_y=1241&top_left_x=825)

## PROBLEM 4

Solve the equation

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

## PROBLEM 5

A deck of $2 n+1$ cards consists of a joker and, for each number between 1 and $n$ inclusive, two cards marked with that number. The $2 n+1$ 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?