[/quote]
,
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
is an oblique strophoid.
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
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 
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)
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)
are taken such that the point 
.
and 
,
and 
.
.
Y by Adventure10, Mango247
Let 
be a positive integer, and let 
be positive integers with 
, such that 
are 
pairwise distinct subsets of the set 
. It is known that 
are pairwise distinct, 
, and runs over all nonempty subsets of . Find all possible values of .
be a positive integer, and let 
be positive integers with 
, such that 
are 
pairwise distinct subsets of the set 
. It is known that 
are pairwise distinct, 
, and runs over all nonempty subsets of . Find all possible values of .
:
,
.
subsets of 
sets of the form 
.
.
(the empty set), and let 
be any permutation of the nonempty subsets of 
.
,
.
.
be the operation that for each set 
then remove it, and if 
insert 
into the set 
.
,
.
with 
being distinct from all 
.
.
and seeing that since 
,
,
when 
(for 
,
so 
is impossible). Thus we can find such a set 
,
.
.
,
by 
.
is nonempty, and thus must be representable in this form. Then note that 
.
.
then
which implies that 
;
into itself with 
,
.
is even. Since 
Define 
and 
The condition implies that for all 
except for 
we have that 
As both are integer polynomials, we know that 
for 
,
Consider the polynomial ![\[ q(x) = 2^\ell(p_A(x) + p_B(x)) - (|A| + |B|) \prod_{i = 1}^\ell (1+x_i). \]](http://latex.artofproblemsolving.com/c/9/8/c98f869744d8b3b074586f792548dc4e804ef314.png)
As this polynomial vanishes on 
and has low degree, it must be identically 
.
as desired.
