KMO 2012

$a,b,c$ are positive numbers such that $a^2 + b^2 + c^2 = 2abc + 1$ . Find the maximum value of $$(a-2bc)(b-2ca)(c-2ab)$$

$p >3$ is a prime number such that $p | 2^{p-1} -1$ and $p \not | 2^x - 1$ for $x = 1, 2, \cdots , p-2$ . Let $p = 2k+3$ . Now we define sequence $\{ a_n \}$ as $$a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 )$$ Prove that there exist $2k$ consecutive terms of sequence $a_{x+1} , a_{x+2} , \cdots , a_{x+2k}$ such that for all $1 \le i < j \le 2k$ , $a_{x+i} \not \equiv a_{x+j} \ (mod \ p)$ .

Let $w$ be the incircle of triangle $ABC$ . Segments $BC, CA$ meet with $w$ at points $D, E$ . A line passing through $B$ and parallel to $DE$ meets $w$ at $F$ and $G$ . ( $F$ is nearer to $B$ than $G$ .) Line $CG$ meets $w$ at $H ( \ne G )$ . A line passing through $G$ and parallel to $EH$ meets with line $AC$ at $I$ . Line $IF$ meets with circle $w$ at $J (\ne F )$ . Lines $CJ$ and $EG$ meets at $K$ . Let $l$ be the line passing through $K$ and parallel to $JD$ . Prove that $l, IF, ED$ meet at one point.

Let $\{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \}$ . Find the maximum value of $$\sum_{n=1}^{10}(na_n ^2 - n^2 a_n )$$

Let $p \equiv 3 \pmod{4}$ be a prime. Define $T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \}$ . For arbitrary subset $S ( \ne \emptyset ) \subset T$ , prove that there exist subset $A \subset S$ satisfying following conditions: (a) $(x_i , y_i ) \in A ( 1 \le i \le 3)$ then $p \not | x_1 + x_2 - y_3$ or $p \not | y_1 + y_2 + x_3$ . (b) $8 n(A) > n(S)$