AoPS Academy
such that 
is a divisor of 
.
be a positive integer, 
, and 
. Prove that 
be an acute triangle with circumcenter 
. Draw altitude 
, with 
on side 
. The parallel line to 
passing through intersects line 
at point 
. Prove that point and the midpoints of sides 
and are collinear.
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 such that Claire can choose the initial numbers such that William cannot attain his goal in less than 
moves., the angle 
is obtuse and 
The line through 
and perpendicular to 
intersects line in 
. Let 
be the midpoint of . Prove that 
.
, the angle is obtuse and The line through and perpendicular to intersects line in . Let be the midpoint of . Prove that .
to intersects 
at point 
. Then 
and 
. Therefore is bisector of triangle 
. So 
is harmonic. And 
, consequently 
is bisector of triangle 
. Hence
.
