AoPS Academy
be a set of 
distinct real numbers. Let 
be the set of numbers that occur as averages of two distinct. For a given 
, what is the smallest possible number of elements in ?
be a convex quadrilateral inscribed in a circle and satisfying 
. Points 
and 
are chosen on sides 
and 
such that 
and 
. Prove that 
.
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.
be normed vector spaces, where 
is a Banach space, and let 
.![\[
X = \left\{ x \in E : \sup_{n \geq 1} \|A_n x\| < +\infty \right\}
\]](http://latex.artofproblemsolving.com/1/4/f/14f368e30ebb290aa9b803da77e96afe63b0c7c0.png)
is either an empty set or second category.
![\[
\frac{(2n)!}{n! \cdot (n+1)!}
\]](http://latex.artofproblemsolving.com/b/9/7/b974bd023a1bf175e6dd0561e4a3a79a55ff7098.png)
is always an integer. Prove that for every 
, there are infinitely many values of 
such that![\[
\frac{(2n)!}{n! \cdot (n+k)!}
\]](http://latex.artofproblemsolving.com/5/f/9/5f9fb4566dfc29e19f658402b4f0bb702441ee62.png)
is an integer.
, show that![\[
\sum_{i=1}^{n} \tan^{-1} r_i = \tan^{-1} \frac{a_1 - a_3 + a_5 - \cdots}{a_0 - a_2 + a_4 - \cdots}
\]](http://latex.artofproblemsolving.com/8/1/2/812c7c690b2237f45bc7453ff2b4160fb72c7b0d.png)
and![\[
\sum_{i=1}^{n} \tanh^{-1} r_i = \tanh^{-1} \frac{a_1 + a_3 + a_5 + \cdots}{a_0 + a_2 + a_4 + \cdots}.
\]](http://latex.artofproblemsolving.com/6/1/9/61962993a2c34e1c0ce961d7057127ae75876ac2.png)
![$f : [0,1] \to \mathbb{R}$](http://latex.artofproblemsolving.com/4/9/b/49bb7defda1cef96adf247814e557ab43137c8a6.png)
continuous such that 
, 
and 
.![$P \in \mathbb{Q}[X]$](http://latex.artofproblemsolving.com/7/9/0/790ff8362f092209912705870106322c0214bfec.png)
such that :![\[ \forall x \in [0,1] : |f(x)-x^{m}P(x)| \leq u \]](http://latex.artofproblemsolving.com/1/6/6/16689ab8bef19c9ab8fa71b1c6e8d27d85fe6a3f.png)
![$(P_{n})_{n\geq 1} \in \mathbb{Q}[X]^{\mathbb{N}}$](http://latex.artofproblemsolving.com/a/c/e/ace9070d3d02a04bf28729b20a10c6496b0ff966.png)
such that 
for all .
and an extractrice 
such that :![\[ \forall x \in [0,1] , n \geq 1, |P_{n}(x)-S_{\phi(n)}(x)| \leq \frac{1}{n} \]](http://latex.artofproblemsolving.com/9/d/0/9d05f2b5d7de79074d74c30dddfd9a3bcb06539e.png)
there is an extractrice such that
converge uniformely to 
in ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
