[/quote]
,
be a positive integer. Show that any polynomial ![$Q\in\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/e/8/2/e8286b6f2b9790f39c88849a1415d86f85c0b92c.png)
with all positive coefficients such that
is a perfect square for any 
must be a constant polynomial.
be an acute triangle and let 
be the foot of the 
-
be the midpoint of 
.
with diameter 
meets 
at 
,
with diameter 
meets 
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)
of degree 
having 
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)
be a circle inscribed inside a rhombus 
.
be variable points on 
is always the tangent line to 
is the same.
are taken such that the point 
.
and 
,
and 
.
.
.
and 
be intersection points of lines 
and 
with circumcircle 
,
and 
with side 
.
,
with 
such that 
divides 
for all positive integers 
with 
.
such that 
Follows for all reals 
Y by
Let 
be an acute triangle with circumradius 
. Let 
be the midpoint of 
and 
the midpoint of 
. The perpendicular to 
through and the perpendicular to through 
intersect at 
. Prove that 
.
be an acute triangle with circumradius 
. Let 
be the midpoint of 
and 
the midpoint of 
. The perpendicular to 
through and the perpendicular to through 
intersect at 
. Prove that 
.This post has been edited 1 time. Last edited by emi3.141592, Sep 29, 2024, 7:54 PM
Reason: Added source
Reason: Added source
,
is cyclic. This implies 
,
is the circumcircle of 
is congruent to 
and 
(note also that 
is cyclic, hence 
are both the diameters of 
)
such that 
and 
.
is a middle segment of 
then 
.
.
.
then 
is cyclic.
then as a consequence of inscribed angles 
.
.
.
then 
.
