AoPS Academy
with 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from to 
. Circumcircle 
cuts circumcircle 
at 
. Let 
be antipode of wrt . Prove that 
are cyclic.
. which are not perfect squares. Calculate the 
-th term of the sequence.
, let be the midpoint of side and be the midpoint of segment 
. Lines 
and intersect at . Show that 
.
such that 
is a square perfect.
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
be the altitude and 
be the intersection point of the altitudes of triangle . Point 
is symmetric to with respect to . The circumcircle of triangle 
intersects at points and 
. Prove that 
.
![$f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$](http://latex.artofproblemsolving.com/8/b/d/8bdc04d75d0654da65463301f6fd0b0af4160b2b.png)
a polynomial whose roots 
are in geometric progression with ration 
. Find 
.
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
, let 
and let 
. The value of 
is:
be a quadrilateral with 
, 
, 
and 
. The measure of 
(in degrees) is:
and 
Prove that
Let 
Prove that
satisfying:
