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.
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 
+1 w
If 
be positive real numbers, show that:
. Find .
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 
.![$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.
, every two opposite sides are equal. Let 
denote the point of intersection of with the perpendicular bisector of 
. Define 
and 
similarly. Prove that , , and are collinear.
is the midpoint of and 
are the orthogonal projections of 
onto 
From 
we get 
On the other hand, from 
and 
we get
into 
yields 
we conclude that 
are collinear.
be the midpoint of and let the perpendiculars from and to the line meet it at points 
, respectively. Notice that 
since we are taking projections along , the 
factor cancels out. Notice that 
and 
. Dividing the two relations yields that 
. Hence, multiplying similar relations gives that 
and by the Menelaus' theorem, the points are collinear. and radius 
and analogous ones for 
are coaxial. Any proof along these lines? 
that satisfies 
for all real 
