and incircle touching 
.
be the antipode of 
in the circumcircle of 
such that 
.
,
be the midpoint of 
.
x
to 
row and 
column get symbolized by square 
. Subset 
belong to 
with 
or 
are not belong to 
denote its incentre and circumcentre respectively. Let 
be the circle through 
and 
which is tangent to the incircle of the triangle 
and 
are defined similarly. The circles 
distinct from 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.
defined as follows:![\[
a_1 = \frac{\sqrt{6}}{3},
\quad
a_{n+1} = a_n + \frac{1}{3a_n},
\quad
\forall n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/8/9/2/892a2676214a3c837d6b18a10a4f6f83bc099f74.png)
![\[
0 \le a_n \sqrt{6} - 2\sqrt{n}
\le
\frac{1}{4\sqrt{n}}
\Bigl(
1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}
\Bigr),
\quad
\forall n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/f/0/8/f0824edd96ddfeaa107d729466134812ad7ea777.png)
b,For each 
,![\[
b_n = \frac{3a_n^2 - 2n - 1}{\ln(n+1)}.
\]](http://latex.artofproblemsolving.com/a/9/3/a937b379140df5aa51d7ede1a685d263c3229e96.png)
Compute the limit 
,
and Moivre's formula.
.
![\[
I = \int_{0}^{\frac{1}{\sqrt{3}}} \frac{u \arctan(u)}{(1 - u^2) \sqrt{1 - 2 u^2}} \, du
\]](http://latex.artofproblemsolving.com/b/c/5/bc5d60ff6679870b47776c401658e1ba62af75aa.png)
![\[
\text{Find: }
10. \sum_{n=1}^\infty \sum_{m=-\infty}^\infty \frac{1}{n^p m^2 (m^2 + 1)^3 (n+1)^q}, \quad 2 \leq p, q \in \mathbb{Z},
\]](http://latex.artofproblemsolving.com/f/3/3/f3360b85ce092804a9de263cac49aed025ea8cee.png)
![\[
11. \sum_{n=1}^\infty \sum_{m=-\infty}^\infty \frac{(-1)^{n+m}}{n^p m^2 (m^2 + 1)^3 (n+1)^q}, \quad 2 \leq p, q \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/9/4/5/945aaa2340a4fb314e5eb218ba5286eb054c1b6f.png)
![\[12.
\int_{0}^{\frac{\pi}{4}} \frac{\sin x}{\cos(2x) + 2} \tan^{-1}\left(\frac{\cos x \cot(2x)}{\sqrt{2}}\right) dx
= \frac{5\pi^2}{48\sqrt{2}} - \frac{\pi}{4\sqrt{2}} \cos^{-1}\left(\frac{1}{3}\right).
\]](http://latex.artofproblemsolving.com/b/1/3/b13a5b1c537926c8323170d3d533b0bc71bb06a1.png)
-
![\[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]](http://latex.artofproblemsolving.com/b/7/2/b72b5e302cecdf2649c43a078aea3aa9a88ef2cc.png)
find 
and 
Y by starchan
Let 
and 
be the altitudes of acute triangle 
. Let 
be the orthocenter of and 
be the midpoint of side 
. The points of intersection of the midperpendicular line to with segments and are denoted by 
and 
respectively. The point 
is the orthocenter of triangle 
. Prove that belongs to the median 
.
(Bohdan Zheliabovskyi)
and 
be the altitudes of acute triangle 
. Let 
be the orthocenter of and 
be the midpoint of side 
. The points of intersection of the midperpendicular line to with segments and are denoted by 
and 
respectively. The point 
is the orthocenter of triangle 
. Prove that belongs to the median 
.(Bohdan Zheliabovskyi)
denote the centre of spiral similarity 
sending segment 
to 
and 
.
respectively since ![\[\angle BFK = \angle CEL = \angle BMK = \angle BML = 90^{\circ}\]](http://latex.artofproblemsolving.com/d/1/a/d1a959589d9dc38a5651a015ed9e57bf48c833b7.png)
Now note that 
has angles equal to 
.
,
to 
be any point. Then 
are collinear.
so this is trivial.
and 
.
,
are collinear. Therefore it suffices to show that 
.
