[/quote]
,
with 
and 
.
?
of positive integers such that
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
be the foot of the 
be the midpoint of 
with diameter 
at 
,
with diameter 
at 
.
are concyclic.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
Y by Rounak_iitr
Let 
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
be an acute triangle with an altitude 
. Let 
be the orthocenter of triangle . The circle 
passes through the points 
and 
, and touches the line 
. Let 
be the diameter of . The lines~
and 
intersect for the second time at points 
and 
, respectively. The lines 
and 
intersect at the point~
. Prove that 
.
are cyclic.
and 
are cyclic.
a desired
and 
are similar
and 
are also similar, proving the equality.
and 
then 
