AoPS Academy
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.
and 
, where is not the zero polynomial, we define 
to be the unique polynomial 
such that we can write 
, where 
is a polynomial such that either 
or the degree of is less than the degree of . Find all polynomials 
with real coefficients such that 
be a triangle.Find the maximum of 
is a cyclic quadrilateral. 
is the midpoint of 
. The diagonals meet at 
. The circle through which touches at meets again at 
and 
again at . The point 
on is such that 
. The line through parallel to meets at 
. Show that 
.
is a cyclic quadrilateral. is the midpoint of . The diagonals meet at . The circle through which touches at meets again at and again at . The point on is such that . The line through parallel to meets at . Show that .
and 
.