1999 ELMO

ELMO 1999

July 1, 1999

Experimental Lincoln Math Olympiad

1. In nonisosceles triangle $ABC$, let 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
$QX_B = QB = QC = QX_C$.

2. 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 $(0,0)$.

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

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

Time: 4 hours.
Each problem is worth 7 points.