AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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: 
. Prove that: 
be a positive integer and let 
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 around the circle such that the product of any two neighbors is of the form 
for some positive integer 
.
be integer numbers 
satisfy that:
. Prove that: are even
is a perfect square number
can’t be any power 
of a positive integer number
such that for all 
, we have![\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]](http://latex.artofproblemsolving.com/a/a/1/aa1bce053788a3efa59819407b41cfe472f544d0.png)
such that
for all 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
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 
. Prove that the points 
and lie on the same line.
be a triangle. Circle 
passes through 
, meets segments 
and 
again at points 
and 
respectively, and intersects segment 
at 
and 
such that lies between 
and . The tangent to circle 
at and the tangent to circle 
at meet at point 
. Suppose that points and are distinct. Prove that line 
is parallel to .
such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
, then we get the relation
. Solve for s.
,
such that:
such that 
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 
.
