ELMO 1999

In nonisosceles triangle $ABC$ the excenters of the triangle opposite $B$ and $C$ be $X_B$ and $X_C$ , respectively. Let the external angle bisector of $A$ intersect the circumcircle of $\triangle ABC$ again at $Q$ . Prove that $QX_B = QB = QC = QX_C$ .

Mr. Fat moves around on the lattice points according to the following rules: From point $(x,y)$ he may move to any of the points $(y,x)$ , $(3x,-2y)$ , $(-2x,3y)$ , $(x+1,y+4)$ and $(x-1,y-4)$ . Show that if he starts at $(0,1)$ he can never get to $(0,0)$ .

Prove that $$2^6 \frac{abcd+1}{(a+b+c+d)^2} \le a^2+b^2+c^2+d^2+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}$$ for $a,b,c,d > 0$ .

Let $a_1, a_2, a_3, \cdots$ be an infinite sequence of real numbers. Prove that there exists a increasing sequence $j_1, j_2, j_3, \cdots$ of positive integers such that the sequence $a_{j_1}, a_{j_2}, a_{j_3}, \cdots$ is either nondecreasing or nonincreasing.