KMO 2008

Hexagon $ABCDEF$ is inscribed in a circle $O$ . Let $BD \cap CF = G, AC \cap BE = H, AD \cap CE = I$ Following conditions are satisfied. $BD \perp CF , CI=AI$ Prove that $CH=AH+DE$ is equivalent to $GH \times BD = BC \times DE$

Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b \mid f(a)+f(b)$ .

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$ .

(1) For every $m\ge2$ , prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$ .

(2) For a certain $m\ge2$ , let $a, b$ be positive integers that satisfy $2^a\in A_m$ , $2^b+1\in A_m$ . Let $a_0, b_0$ be the least such pair $a, b$ . Find the relation between $a_0$ and $b_0$ .

Quadrilateral $ABCD$ is inscribed in a circle $O$ . Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$ , $R\in AD, ER\perp AD$ , $EP\cap AD=Q, ER\cap BC=S$ Let $K$ be the midpoint of $QS$ Prove that $E, K, O$ are collinear.

There is $n\times n$ chessboard. Each square has a number between $0$ and $k$ . There is a button for each row and column, which increases the number of $n$ numbers of the row or column the button represents(if the number of the square is $k$ , then it becomes $0$ ). If certain button is pressed, call it 'operation.'

And we have a chessboard which is filled with 0(for all squares). After some 'operation's, the numbers of squares are different now. Prove that we can make all of the number $0$ within $kn$ 'operation's.