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 real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
be a triangle with 
and 
, and let 
be the midpoint of 
. Let 
be a variable point of the circumcircle of , and let 
be the point on the segment 
such that 
. It is given that the line through and perpendicular to intersects the line 
at a unique point 
. lies on this circle for all possible positions of .
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
be a parallelogram Let 
be a point in its interior such that 
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
such that ![\[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]](http://latex.artofproblemsolving.com/4/4/2/4423651bb829b92debd65a23f6cee3d4d4b8530f.png)
holds for all reals 
and 
.
. At a maths camp, there are 
children, each of whom selects
tables, with children tables, with children
and 
. Prove that
Let and 
. Prove that
and 
. Prove that
Where 
and 
. Prove that
be an acute scalene triangle with circumcenter . Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through tangent to 
respectively. Let 
be the intersection of 
and 
. Let be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
![\[ p = \frac{2ab}{a + b} \]](http://latex.artofproblemsolving.com/0/f/4/0f4297ec59fe97c92fda75af4cc835be6b9a2507.png)
, ![\[ p, a, b \in \mathbb{C} \]](http://latex.artofproblemsolving.com/b/4/4/b4499970ac49d770a810d7345fedef7773cd03f5.png)
be a prime number. Prove that there exists a natural number 
such that![\[
p \mid m^n - n.
\]](http://latex.artofproblemsolving.com/3/a/3/3a38b299f0328d176286d65e02b92e9918e8109d.png)
just pick 
, otherwise 
is coprime to 
. By Chinese Remainder Theorem there exists an such that 
and 
, pick this , we see that both 
and leave remainder 
modulo , so we are done.
Prove that
Prove that
