CAPS 2012

Given a positive integer $n$ , let $\tau(n)$ denote the number of positive divisors of $n$ and $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are relatively prime to $n$ . Find all $n$ for which one of the three numbers $n,\tau(n), \varphi(n)$ is the arithmetic mean of the other two.

City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$ . A street cleaning car starts from point $W$ . The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$ -meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.)

Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$ . The line, perpendicuar to $CP$ and passing through $C$ , intersects $AP$ , $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$ .