AoPS Academy
be an acute triangle and 
its orthocenter. Let 
be a point on the parallel through 
to 
such that 
. Define 
and 
as points on the parallels through 
to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If 
is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
, 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)
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
converge?
and 
.
with multiplication of matrixes operation is making an isomorphic-group structure with 
.
. bijective, and with converge.
converge?
be an even positive integer. Let 
be a monic, real polynomial of degree 
; that is to say, 
for some real coefficients 
. Suppose that 
for all integers 
such that 
. Find all other real numbers 
for which 
.
be a real-valued function with 
derivatives at each point of 
. Show that for each pair of real numbers 
, 
, 
, such that
there is a number 
in the open interval 
for which
and see for myself that it was symetric wrt variables A B C)
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
,
is equilateral.
the proof is easy. Without trig, there are few proofs I have seen a simple search in google will give a few interesting proofs..
