[/quote]
,
with 
and 
.
?
of positive integers such that
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 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)
Y by
The sequence 
of integers is such that each number occurs in no more than 
times. Prove that there are two strictly increasing sequences of indices 
and 
are such that every positive integer from the set 
occurs exactly in one of these two sequences, and for each 
is true the condition 
.
(Anton Trygub)
of integers is such that each number occurs in no more than 
times. Prove that there are two strictly increasing sequences of indices 
and 
are such that every positive integer from the set 
occurs exactly in one of these two sequences, and for each 
is true the condition 
.
(Anton Trygub)
works. So assume that the statement holds for 
.
be a sequence of integers.
times. Then we let all 
'
'
only appear at most 
times. Then let 
.
.
with 
terms. By induction, there exist 
and 
that satisfy the condition.
and 
.
,
,
