FKMO 2013

For a triangle $\triangle ABC (\angle B > \angle C)$ , $D$ is a point on $AC$ satisfying $\angle ABD = \angle C$ . Let $I$ be the incenter of $\triangle ABC$ , and circumcircle of $\triangle CDI$ meets $AI$ at $E ( \ne I )$ . The line passing $E$ and parallel to $AB$ meets the line $BD$ at $P$ . Let $J$ be the incenter of $\triangle ABD$ , and $A'$ be the point such that $AI = IA'$ . Let $Q$ be the intersection point of $JP$ and $A'C$ . Prove that $QJ = QA'$ .

Find all functions $f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions. (a) $f(x) \ge 0$ for all $x \in \mathbb{R}$ . (b) For $a, b, c, d \in \mathbb{R}$ with $ab + bc + cd = 0$ , equality $f(a-b) + f(c-d) = f(a) + f(b+c) + f(d)$ holds.

For a positive integer $n \ge 2$ , define set $T = \{ (i,j) | 1 \le i < j \le n , i | j \}$ . For nonnegative real numbers $x_1 , x_2 , \cdots , x_n$ with $x_1 + x_2 + \cdots + x_n = 1$ , find the maximum value of $$\sum_{(i,j) \in T} x_i x_j$$ in terms of $n$ .

For a triangle $ABC$ , let $B_1 ,C_1$ be the excenters of $B, C$ . Line $B_1 C_1$ meets with the circumcircle of $\triangle ABC$ at point $D (\ne A)$ . $E$ is the point which satisfies $B_1 E \bot CA$ and $C_1 E \bot AB$ . Let $w$ be the circumcircle of $\triangle ADE$ . The tangent to the circle $w$ at $D$ meets $AE$ at $F$ . $G , H$ are the points on $AE, w$ such that $DGH \bot AE$ . The circumcircle of $\triangle HGF$ meets $w$ at point $I ( \ne H )$ , and $J$ be the foot of perpendicular from $D$ to $AH$ . Prove that $AI$ passes the midpoint of $DJ$ .

Two coprime positive integers $a, b$ are given. Integer sequence $\{ a_n \}, \{b_n \}$ satisties $$(a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2$$ Find all prime numbers $p$ such that there exist positive integer $n \le p$ satisfying $p | b_n$ .

For any permutation $f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \}$ , and define $$A = \{ i | i > f(i) \}$$ $$B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \}$$ $$C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \}$$ $$D = \{ (i, j) | i< j \ and \ f(i) > f(j)\}$$ Prove that $|A| + 2|B| + |C| = |D|$ .