CAPS 2009

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$\Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1$$ for all $x,y\in\mathbb{R}^+$ .

For positive integers $a$ and $k$ , define the sequence $a_1,a_2,\ldots$ by $$a_1=a, \text{and} a_{n+1}=a_n+k\cdot\varrho(a_n) \text{for } n=1,2,\ldots$$ where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$ ). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.

Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$ . A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$ , respectively. Let $\omega'$ denote the incircle of triangle $ADE$ . The tangent from $D$ to $\omega$ (different from line $AB$ ) and the tangent from $E$ to $\omega$ (different from line $AC$ ) meet at point $P$ . The tangent from $B$ to $\omega'$ (different from line $AB$ ) and the tangent from $C$ to $\omega'$ (different from line $AC$ ) meet at point $Q$ . Prove that, independent of the choice of $\ell$ , there is a fixed point that line $PQ$ always passes through.

Given a circle, let $AB$ be a chord that is not a diameter, and let $C$ be a point on the longer arc $AB$ . Let $K$ and $L$ denote the reflections of $A$ and $B$ , respectively, about lines $BC$ and $AC$ , respectively. Prove that the distance between the midpoint of $AB$ and the midpoint of $KL$ is independent of the choice of $C$ .

The $n$ -tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following: list $1\le a_1<a_2<\cdots < a_n\le 50$ (ii) for each $n$ -tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$ -tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that $$mb_i=c_i^{a_i} \text{for } i=1,2,\ldots,n.$$ [/list]Prove that $n\le 16$ and determine the number of $n$ -tuples $(a_1,a_2,\ldots,a_n$ ) satisfying these conditions for $n=16$ .

Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane: $$G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.$$ Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.