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.
, where 
and 
.
satisfies the following property:![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](http://latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
and for all 
.
be a real numbers such that 
Then find maximum value of 
segments 
and 
are parallel. Angle bisectors of 
and 
meet at 
. Angle bisectors of 
and 
meet at 
. Prove that 
is cyclic
such that 
is even , 
is odd and 
is divisible by 
.Prove that there exists a prime number 
such that is divisible by 
and 
, then find the maximum value of ![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
such that there exists a prime number , such that 
is a power of 
. is a number of the form 
where is a positive integer.
be an acute triangle with orthocenter 
and circumcenter 
. Let 
and 
be points on the line segments 
and respectively such that 
is a parallelogram. Prove that 
.
, let 
be the midpoint of side 
. Let the line perpendicular to at point 
intersect 
at 
. If 
is tangent to at 
, find 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line. and are positive integers, we say that almost divides if divides at least one of 
and 
. We call a positive integer almost prime if the following holds: for any positive integers 
such that almost divides 
, we have that almost divides at least one of and . Determine all almost prime numbers.
is an odd number, let 
be a positive integer. If 
then 
such that 
. Find the minimum: 
with bases and 
and lateral sides and , their midpoints are denoted 
, respectively. Find the lengths of the bases and , if 
, 
, 
, 
.
with bases and and lateral sides and , their midpoints are denoted , respectively. Find the lengths of the bases and , if , , , .
and 
Prove that
.
