.
,
on side 
passing through 
at point 
.
be a positive integer. Claire writes 
distinct positive real numbers 
in a row on a blackboard. In a 
William can erase a number 
and replace it with either 
or 
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
independent of 
such that William can always reach his goal in at most 
moves.
independent of 
moves.
and 
,
to be the unique polynomial 
such that we can write 
,
is a polynomial such that either 
or the degree of 
with real coefficients such that 
Y by
Let 
be a triangle, and let 
and 
be points on segment 
symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line 
at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
be a triangle, and let 
and 
be points on segment 
symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line 
at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
.
.
.
cyclic gives us 
is a point on the extension of the ray 
implies that triangles 
and 
similar. We get 
=
.
=
and 
.
and 
are similar. Implies that 
.
