CANADA_MO/md/en-exam1995.md

Canadian Mathematical Olympiad

1995

Problem 1

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

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

PROBLEM 2

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

$$a^a{b}^b{c}^c\ge (abc{)}^{\frac{a+b+c}{3}}a^\{a\}\; b^\{b\}\; c^\{c\}\; \backslash geq(a\; b\; c)^\{\backslash frac\{a+b+c\}\{3\}\}$$

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 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

$$x_1^3+x_2^3+\cdots +x_n^3=y^{3k+2}x\_\{1\}^\{3\}+x\_\{2\}^\{3\}+\backslash cdots+x\_\{n\}^\{3\}=y^\{3\; k+2\}$$

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

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

$$u_1=f(1),\text{ and }{u}_n=f\left(u_{n-1}\right)\text{ for all }n>1u\_\{1\}=f(1),\; \backslash text\; \{\; and\; \}\; u\_\{n\}=f\backslash left(u\_\{n-1\}\backslash right)\; \backslash text\; \{\; for\; all\; \}\; n>1$$

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