AoPS Academy
be positive real numbers such that : 
. Prove that : 
be an acute triangle with 
. Let 
be the centroid of triangle 
and let 
be the foot of the perpendicular from 
to side 
. The median 
intersects the circumcircle 
of triangle at a second point 
. Let 
be the point where intersects . The line 
intersects the circle at a point 
, such that lies between and . Prove that the points 
and lie on a circle.
be non-negative real numbers such that 
Prove that
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and be points on the line segments 
and respectively such that 
is a parallelogram. Prove that 
.
be a trapezoid. By 
we denote the angle bisector of angle 
. Let 
and 
. Prove that 
is cyclic.
be real positive number satisfy that: 
. Prove that: 
, where 
integers and 
is a positive integer. Suppose that the integers
satisfy: 
are divisible by k. Prove that:
is divisible by k
such that 
and
.
and 
are primes, then can 
be a perfect square?
and 
Prove that
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
![[asy]
size(150);
draw((-1,0)--(3,0)); // ground line
draw(circle((0,1),1)); // left big circle
draw(circle((2,1),1)); // right big circle
draw(circle((1,0.25),0.25)); // small circle in between
[/asy]](http://latex.artofproblemsolving.com/d/0/d/d0d9cb2706e9b913e2051e224b8e64537c7b8332.png)
;