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.
with circumcircle 
. 
is the midpoint of segment 
and 
is the midpoint of 
of (the one that doesn't contain 
). 
and 
are points on such that 
. Assume there exists point 
on segment such that circumcircle of triangle 
is tangent to . Let 
be the circumcircle of triangle 
. Line 
meets for the second time at 
. Let 
be a point on such that 
be a circle thaat passes through 
, and tangents to and 
be a circle that passes through 
and tangents to . Prove that circle with center and radias 
is tangent to circles 
and .
triangle with 
. We take , point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point . If 
, find 
= 2∠AMP.
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 
.
.
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)
with area 
and a point 
in the same plane, determine the area of polygon 
where 
is the image of 
around 
by 
