CWMO 2010

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that**(a)** $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$ ;(b) $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$ .

$AB$ is a diameter of a circle with center $O$ . Let $C$ and $D$ be two different points on the circle on the same side of $AB$ , and the lines tangent to the circle at points $C$ and $D$ meet at $E$ . Segments $AD$ and $BC$ meet at $F$ . Lines $EF$ and $AB$ meet at $M$ . Prove that $E,C,M$ and $D$ are concyclic.

Determine all possible values of positive integer $n$ , such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$ , with $|A_i \cap A_j| \not= 1$ for all $i \not= j$ .

Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:

(1) $\sum_{i=1}^{n} (a_i + b_i) = 1$ ;

(2) $\sum_{i=1}^{n} i(a_i - b_i) = 0$ ;

(3) $\sum_{i=1}^{n} i^2(a_i + b_i) = 10$ .

Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$ .

Let $k$ be an integer and $k > 1$ . Define a sequence $\{a_n\}$ as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$ for $n = 1,2,...$ .

Determine, with proof, all possible $k$ for which there exist non-negative integers $l,m (l \not= m)$ and positive integers $p,q$ such that $a_l + ka_p = a_m + ka_q$ .

$\Delta ABC$ is a right-angled triangle, $\angle C = 90^{\circ}$ . Draw a circle centered at $B$ with radius $BC$ . Let $D$ be a point on the side $AC$ , and $DE$ is tangent to the circle at $E$ . The line through $C$ perpendicular to $AB$ meets line $BE$ at $F$ . Line $AF$ meets $DE$ at point $G$ . The line through $A$ parallel to $BG$ meets $DE$ at $H$ . Prove that $GE = GH$ .

There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$ , if at least one of player $A$ 's losers is not a $B$ 's loser.

Determine, with proof, all possible values of $n$ , such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.

Determine all possible values of integer $k$ for which there exist positive integers $a$ and $b$ such that $\dfrac{b+1}{a} + \dfrac{a+1}{b} = k$ .