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and 
be positive real numbers such that 
. Prove that![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
, ![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
and 
Prove that
Let and 
Prove that
. 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:
. We have 
, because 
. So, 
and we are done.
and 
Prove that
satisfying:
are the reflections of vertices 
about the opposite sidelines of triangle 
and 
have a common point.
and 
meet at 
.
which clearly sum to 180 so we are done.