with 
be given . On the extension of the segment 
beyond 
,
is chosen arbitrarily. Let the line through 
at the point 
.
be the intersection of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
and 
Prove that
Let 
Prove that
.
-
,
.
and 
.
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 
Prove that
Let 
Prove that
be the altitude and 
be the intersection point of the altitudes of triangle 
is symmetric to 
intersects 
.
.
![$f\in\mathbb{C}[X], \ f=aX^3+bX^2+cX+d, \ a,b,c,d\in\mathbb{R}^*$](http://latex.artofproblemsolving.com/8/b/d/8bdc04d75d0654da65463301f6fd0b0af4160b2b.png)
a polynomial whose roots 
are in geometric progression with ration 
.
.
satisfying:
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
,
and let 
.
is:
,
,
and 
.
(in degrees) is:
Y by Adventure10, Mango247
There are some cities in a country; one of them is the capital. For any two cities 
and 
there is a direct flight from to and a direct flight from to , both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.
and 
there is a direct flight from to and a direct flight from to , both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.
as the price of the flight from 
to 
.
.
.
and 
.
.
and 
.
is constant for any two cities 
,
.
be the cost of a round trip landing in every city exactly once.
,
,
