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 the circumcenter and 
the orthocenter of an acute triangle 
Prove that the area of one of the triangles 
and 
is equal to the sum of the areas of the other two.
with any line through the centroid 
and I have no clue as to why that is true. Plz help
be positive integers, and define![\[ n = a + b + c + d + e + f. \]](http://latex.artofproblemsolving.com/9/d/d/9ddfecfa5800050bc15dcaa48d17d12e431fc358.png)
Suppose that 
divides both of the following expressions:![\[
abc + def \quad \text{and} \quad ab + bc + ca - de - ef - fd.
\]](http://latex.artofproblemsolving.com/c/5/8/c5817937f736a7b00c85e1f92b8e561ba26cf8b0.png)
Prove that is a composite number.
be a triangle with orthocentre . Let 
be the foot of altitudes of 
onto the opposite sides, respectively. Consider 
, the reflection of 
about line 
. Let line 
cut at distinct points 
, and let 
be the orthocenter of 
. Prove that points 
are concyclic.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
such that 
is chosen on the side 
of a triangle with 
in such a way that 
. The incircle of is tangent to 
and at points 
and 
, respectively. Let 
be the incenter of triangle 
. Prove that the line 
intersects the line segment 
at its midpoint.
be an integer. We call a pair of integers 
good if ![\[0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b\]](http://latex.artofproblemsolving.com/e/8/d/e8df562e024b8476d69fdaea15338c27bb6c4eb8.png)
Prove that the number of good pairs is a power of 
.
be an equilateral triangle with incircle . A point on is reflected in the sides of to obtain a new triangle 
. The same point is then reflected over the sides of to obtain another triangle 
. Prove that the circumcircle of is tangent to .
is differentiable and 
Show that for some 
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)
is differentiable with 
If ![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)
then show that 
be the unit circle in the complex plane. Let 
be the map given by 
We define 
and 
for 
The smallest positive integer such that 
is called period of 
Determine the total number of points 
of period 
denote the set of natural numbers, and let 
be nine distinct tuples in 
Show that there are 
distinct elements in the set 
whose product is a perfect cube.
and let 
be positive integers such that 
Prove that 
and determine when equality holds.
is analytic on a closed curve 
(i.e., 
is analytic in a region that contains ). Show that ![\[\int_\gamma \overline{f(z)} f'(z)\,\mathrm dz\]](http://latex.artofproblemsolving.com/c/3/7/c3784659f77e99585032e21fef95bd31a11c23e4.png)
is purely imaginary. (The continuity of 
is taken for granted.)
![\[\sum\limits_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}\]](http://latex.artofproblemsolving.com/2/6/0/26047fb1cf4216f01b301beb9458237025593d06.png)
![\[\sum\limits_{n=0}^{\infty}(-1)^{n}\left(\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\cdots\right)^{2}\]](http://latex.artofproblemsolving.com/b/e/e/beeed44a57c5f75b697062f717687c067523006a.png)
, where
![$ \underbrace{\Big[\tan u\cdot \log^2(\sin u)\Big]_0^{\frac{\pi}{2}}}_{=0}-2\int_0^{\frac{\pi}{2}} \log(\sin u)\, du=\pi \log 2 $](http://latex.artofproblemsolving.com/b/6/8/b68368c76efe19709b4588b79fc7e2a146bdc9a4.png)
and
![$ \Big[-u^2\cot u\Big]_0^{\frac{\pi}{2}}+2\int_0^{\frac{\pi}{2}} u\cot u\, du=\pi \log 2 $](http://latex.artofproblemsolving.com/f/f/8/ff866ee788d9f73f82f92f874efc6d5c01e657ca.png)
.
. And I guess 
.