PAMO 2008

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x+y)\le f(x)+f(y)\le x+y$ for all $x,y\in\mathbb{R}$ .

Let $C_1$ be a circle with centre $O$ , and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$ . Consider a point $T$ on the circle $C_2$ with diameter $OM$ . The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$ .

Let $a,b,c$ be three positive integers such that $a<b<c$ . Consider the the sets $A,B,C$ and $X$ , defined as follows: $A=\{ 1,2,\ldots ,a \}$ , $B=\{a+1,a+2,\ldots,b\}$ , $C=\{b+1,b+2,\ldots ,c\}$ and $X=A\cup B\cup C$ . Determine, in terms of $a,b$ and $c$ , the number of ways of placing the elements of $X$ in three boxes such that there are $x,y$ and $z$ elements in the first, second and third box respectively, knowing that: i) $x\le y\le z$ ; ii) elements of $B$ cannot be put in the first box; iii) elements of $C$ cannot be put in the third box.

Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$ .

A set of positive integers $X$ is called connected if $|X|\ge 2$ and there exist two distinct elements $m$ and $n$ of $X$ such that $m$ is a divisor of $n$ . Determine the number of connected subsets of the set $\{1,2,\ldots,10\}$ .

Prove that for all positive integers $n$ , there exists a positive integer $m$ which is a multiple of $n$ and the sum of the digits of $m$ is equal to $n$ .