CANADA_MO/md/en-exam1992.md

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

$$x(x-z{)}^2+y(y-z{)}^2\ge (x-z)(y-z)(x+y-z)x(x-z)^\{2\}+y(y-z)^\{2\}\; \backslash geq(x-z)(y-z)(x+y-z)$$

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 $P U Q V$.

PROBLEM 4

Solve the equation

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

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?