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.
If 
be positive real numbers, show that:
such that 
and 
are also primes.
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
be a positive integer and let 
be odd primes. Prove that the positive integers 
can be colored in 
colors, such that for any 
of the same color, 
is not divisible by and 
.
be a triangle and let 
be its circumcenter and 
its incenter.
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
be the Gergonne point of the triangle .
is parallel with line 
.
be a polynomial with real coefficients such that 
for all 
. Prove that 
for some real constant 
and 
.
= 2∠AMP.
.
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly times and there exist integers 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
, there exists an integer 
such that 
and 
are both divisible by . How many such labelings exist?
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
or 
. To avoid unfairly favoring specific company, the regulation ensures that if there have three cities 
and 
, with flight routes 
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from and .
, 
and 
denote the number of flight routes operated by companies and , respectively. It is known that, starting from a city, we can arrive any other city through a series of flight routes (not necessary operated by the same company). Find the minimum possible value of 
. is a positive integer. Determine all functions 
such that![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](http://latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
be a fixed integer. Find the least constant such the inequality![\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]](http://latex.artofproblemsolving.com/d/d/e/dde2d1dd0c412d8ba640f1d4c044e2fa2dcccc82.png)
(the sum on the left consists of 
summands). For this constant , characterize the instances of equality.
, let 
be the least prime that does not divide 
, and define 
to be the product of all primes less than . In particular, 
For having 
, define 
. Consider the sequence 
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
. Find all such that 
.
.let 
be a point on 
,different from its midpoint.
is on the extension of segment 
and 
,
and
intersect at 
respectively.prove that 
are concyclic.
.let be a point on ,different from its midpoint. is on the extension of segment and ,andintersect at respectively.prove that are concyclic.
, and can be chosen independently....
.
is , is a point on side (not the midpoint of side ), is a point on the extension of line segment 
, straight line 
intersects at point 
, and straight line intersects at point 
. Prove that if 
, then the points 
, 
, and lie on a circle.
points.
seems wrong with my picture)
is concyclic,
is the polar of 
isn't parallel to 
with 
given. But I can't figure out how to adjust the definition order.