PAMO 2002

Find all functions $f: N_0 \to N_0$ , (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$ .

$\triangle{AOB}$ is a right triangle with $\angle{AOB}=90^{o}$ . $C$ and $D$ are moving on $AO$ and $BO$ respectively such that $AC=BD$ . Show that there is a fixed point $P$ through which the perpendicular bisector of $CD$ always passes.

Prove for every integer $n>0$ , there exists an integer $k>0$ such that $2^nk$ can be written in decimal notation using only digits 1 and 2.

Seven students in a class compare their marks in 12 subjects studied and observe that no two of the students have identical marks in all 12 subjects. Prove that we can choose 6 subjects such that any two of the students have different marks in at least one of these subjects.

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$ , then prove: $$a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1$$