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The three lines AA', BB' and CC' meet on the line IO
WakeUp 44
N
by ihategeo_1969
Source: Romanian Master Of Mathematics 2012
Let 
be a triangle and let 
and 
denote its incentre and circumcentre respectively. Let 
be the circle through 
and 
which is tangent to the incircle of the triangle ; the circles 
and 
are defined similarly. The circles and meet at a point 
distinct from 
; the points 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.
(Russia) Fedor Ivlev
be a triangle and let 
and 
denote its incentre and circumcentre respectively. Let 
be the circle through 
and 
which is tangent to the incircle of the triangle ; the circles 
and 
are defined similarly. The circles and meet at a point 
distinct from 
; the points 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.(Russia) Fedor Ivlev
44 replies
+1 w
Gaussian integral
soruz 3
N
by Mathzeus1024
Exist a method of calculation for 
, with help of 
and Moivre's formula.
, with help of 
and Moivre's formula.
3 replies
Limit conundrum
MetaphysicalWukong 4
N
by MetaphysicalWukong
Source: UNSW
Why is the last statement not true? And how do we know the selected option is true?
4 replies
Finding supremum of a weird function
pokoknyaakuimut 4
N
by MihaiT
Find 
. Easy to guess that the answer is 
, but I haven't found the reason yet. :(
. Easy to guess that the answer is 
, but I haven't found the reason yet. :(
4 replies
Integration Bee Kaizo
Calcul8er 50
N
by Shikhar_
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
50 replies
Putnam 1950 B1
centslordm 2
N
by KAME06
In each of 
houses on a straight street are one or more boys. At what point should all the boys meet so that the sum of the distances that they walk is as small as possible?
houses on a straight street are one or more boys. At what point should all the boys meet so that the sum of the distances that they walk is as small as possible?2 replies
![\[
I = \int_{0}^{\frac{1}{\sqrt{3}}} \frac{u \arctan(u)}{(1 - u^2) \sqrt{1 - 2 u^2}} \, du
\]](http://latex.artofproblemsolving.com/b/c/5/bc5d60ff6679870b47776c401658e1ba62af75aa.png)
Some integrals and sums(series)
Martin.s 1
N
by Entrepreneur
Source: Inspired from silver08
I saw Silver's post, so I thought I'd share some integrals and sums as well.
It's Christmas!!! (or boxing day.)
![\[
\text{Find: }
10. \sum_{n=1}^\infty \sum_{m=-\infty}^\infty \frac{1}{n^p m^2 (m^2 + 1)^3 (n+1)^q}, \quad 2 \leq p, q \in \mathbb{Z},
\]](//latex.artofproblemsolving.com/f/3/3/f3360b85ce092804a9de263cac49aed025ea8cee.png)
![\[
11. \sum_{n=1}^\infty \sum_{m=-\infty}^\infty \frac{(-1)^{n+m}}{n^p m^2 (m^2 + 1)^3 (n+1)^q}, \quad 2 \leq p, q \in \mathbb{Z}.
\]](//latex.artofproblemsolving.com/9/4/5/945aaa2340a4fb314e5eb218ba5286eb054c1b6f.png)
![\[12.
\int_{0}^{\frac{\pi}{4}} \frac{\sin x}{\cos(2x) + 2} \tan^{-1}\left(\frac{\cos x \cot(2x)}{\sqrt{2}}\right) dx
= \frac{5\pi^2}{48\sqrt{2}} - \frac{\pi}{4\sqrt{2}} \cos^{-1}\left(\frac{1}{3}\right).
\]](//latex.artofproblemsolving.com/b/1/3/b13a5b1c537926c8323170d3d533b0bc71bb06a1.png)
It's Christmas!!! (or boxing day.)
![\[
\text{Find: }
10. \sum_{n=1}^\infty \sum_{m=-\infty}^\infty \frac{1}{n^p m^2 (m^2 + 1)^3 (n+1)^q}, \quad 2 \leq p, q \in \mathbb{Z},
\]](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)
1 reply
-
![\[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]](http://latex.artofproblemsolving.com/b/7/2/b72b5e302cecdf2649c43a078aea3aa9a88ef2cc.png)
find 
and 
