# Canadian Mathematical Olympiad 

1995

## Problem 1

Let $f(x)=\frac{9^{x}}{9^{x}+3}$. Evaluate the sum

$$
f\left(\frac{1}{1996}\right)+f\left(\frac{2}{1996}\right)+f\left(\frac{3}{1996}\right)+\cdots+f\left(\frac{1995}{1996}\right)
$$

## PROBLEM 2

Let $a, b$, and $c$ be positive real numbers. Prove that

$$
a^{a} b^{b} c^{c} \geq(a b c)^{\frac{a+b+c}{3}}
$$

## PROBLEM 3

Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than 180 degrees. (See Figure displayed.) Let $C$ be a convex polygon having 5 sides. Suppose that the interior region of C is the union of $q$ quadrilaterals, none of whose interiors intersect one another. Also suppose that $b$ of these quadrilaterals are boomerangs. Show
![](https://cdn.mathpix.com/cropped/2024_11_22_fc9f58e8dd728edf9dbcg-1.jpg?height=281&width=378&top_left_y=1128&top_left_x=1166)
that $q \geq b+\frac{s-2}{2}$.

Problem 4
Let $n$ be a fixed positive integer. Show that for only nonnegative integers $k$, the diophantine equation

$$
x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}=y^{3 k+2}
$$

has infinitely many solutions in positive integers $x_{i}$ and $y$.
Problem 5
Suppose that $u$ is a real parameter with $0<u<1$. Define

$$
f(x)= \begin{cases}0 & \text { if } 0 \leq x \leq u \\ 1-(\sqrt{u x}+\sqrt{(1-u)(1-x)})^{2} & \text { if } u \leq x \leq 1\end{cases}
$$

and define the sequence $\left\{u_{n}\right\}$ recursively as follows:

$$
u_{1}=f(1), \text { and } u_{n}=f\left(u_{n-1}\right) \text { for all } n>1
$$

Show that there exists a positive ineger $k$ for which $u_{k}=0$.