[/quote]
,
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
be real positive numbers such that: 
.
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of 
for some positive integer 
.
be integer numbers 
satisfy that
.
is a perfect square number
can’t be any power 
of a positive integer number
such that:
such that for all 
,![\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]](http://latex.artofproblemsolving.com/a/a/1/aa1bce053788a3efa59819407b41cfe472f544d0.png)
such that 
such that
for all 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
and 
.
be a triangle. Circle 
passes through 
,
and 
again at points 
and 
at 
such that 
and 
at 
at 
.
is parallel to 
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
Y by Adventure10, Mango247
Let 
be an acute angled triangle and let 
be its circumscribed circle. A point 
is given in the interior of the triangle, such that 
, where 
and 
are on and lies on 
while lies on 
. Prove that is on the angle bisector of 
if and only if 
.
be an acute angled triangle and let 
be its circumscribed circle. A point 
is given in the interior of the triangle, such that 
, where 
and 
are on and lies on 
while lies on 
. Prove that is on the angle bisector of 
if and only if 
.
since 
is bisector of 
and is easy to see that: 
since 
since 
and 
