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.
be nonnegative positive integers.
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
be a cyclic quadrilateral with 
and let 
and be the feet of altitudes from 
to the lines 
and 
,.Prove 
bisects 
.
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)
![$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 .
= 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 
.
independent of point 
,
is tangent to 
is determined uniquely by this circle.
