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.
that satisfies 
for all real 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
such that there exists a monic polynomial of degree and with irrational coefficients ( excepts its leading term ) and such that it has distinct irrational roots.
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.
and be natural numbers such that 
is a prime number. What is the remainder of the number when divided by 
?
be a function which is differentiable on 
and satisfies![\[
\lim_{x\to+\infty}\bigl(f'(x)/e^x\bigr)=0.
\]](http://latex.artofproblemsolving.com/a/f/b/afbd725dc4a4cbb1ecfdc1d4923f9a7edcb53397.png)
Prove that![\[
\lim_{x\to+\infty}\bigl(f(x)/e^x\bigr)0.
\]](http://latex.artofproblemsolving.com/8/c/a/8caa7f6cc28759036af097a634f000c6daf6bb79.png)
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
If 
be positive real numbers, show that:
. Find .
be nonnegative positive integers.
be a cyclic quadrilateral with 
and let 
and be the feet of altitudes from 
to the lines 
and 
,.Prove 
bisects 
.![$P\in \mathbb{R}[x] $](http://latex.artofproblemsolving.com/1/1/9/1194a1382b5ae7a9945cf8d9a6281954cf8cdb3d.png)
with even-degree
be an acute triangle with incenter 
, circumcenter 
, and circumcircle 
. Let be the midpoint of 
. Ray 
meets 
at 
. Denote by 
and 
the circumcircles of 
and 
, respectively. Line 
meets at 
and 
, while line 
meets at 
and . Assume that lies inside 
and 
. at and and the tangents to at 
and . Given that 
, prove that these four lines are concurrent on . 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 .
