# Canadian Mathematical Olympiad 

1983

## Problem 1

Find all positive integers $w, x, y$ and $z$ which satisfy $w!=x!+y!+z!$.

## Problem 2

For each real number $r$ let $T_{r}$ be the transformation of the plane that takes the point $(x, y)$ into the point $\left(2^{r} x, r 2^{r} x+2^{r} y\right)$. Let $F$ be the family of all such transformations i.e. $F=\left\{T_{r}: r\right.$ a real number $\}$. Find all curves $y=f(x)$ whose graphs remain unchanged by every transformation in $F$.

## PROBLEM 3

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

Problem 4
Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^{n}-n$.

Problem 5
The geometric mean (G.M.) of a $k$ positive numbers $a_{1}, a_{2}, \ldots, a_{k}$ is defined to be the (positive) $k$-th root of their product. For example, the G.M. of $3,4,18$ is 6 . Show that the G.M. of a set $S$ of $n$ positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of $S$.