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{\noindent {\bf ELMO 1999} \hfill {\bf E}xperimental {\bf L}incoln {\bf M}ath {\bf O}lympiad}

\begin{enumerate}
\item
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 $QX_B = QB = QC = QX_C$.


\item
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 $(0, 0)$.


\item
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 $a$, $b$, $c$, $d > 0$.


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

\end{enumerate}

\vfill

Time: 4 hours.

Each problem is worth 7 points.

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