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, angle 
. Point 
lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
be a parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
. 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:
that satisfies that for any different integers, the product of all the positive differences of these numbers is divisible by 
.
that satisfies that for any different integers, the product of all the positive differences of these numbers is divisible by .
and 
Prove that
Let 
Prove that
satisfying:
