Physical or online

by wimpykid, Apr 30, 2025, 6:49 AM

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Do you think the AoPS print books or the online books are better?

Generating Functions

by greenplanet2050, Apr 29, 2025, 10:42 PM

So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you

function

Transformation of a cross product when multiplied by matrix A

by Math-lover1, Apr 29, 2025, 10:29 PM

I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.

AoPS Volume 2 wrote:

A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$ ] corresponds to a matrix which reverses the direction of the cross product of two vectors.

I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$ , but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?

This post has been edited 1 time. Last edited by Math-lover1, Yesterday at 10:30 PM
Reason: edit

linear algebra

matrix

vector

trigonometric functions

by VivaanKam, Apr 29, 2025, 8:29 PM

Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!

trigonometry

geometry

function

Geometry books

by T.Mousavidin, Apr 29, 2025, 4:25 PM

Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,

projective geometry

geometry

3D geometry

Sequence

by lgx57, Apr 27, 2025, 12:56 PM

$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$ . Find the general term of $\{a_n\}$ .

algebra

Sequences

Inequalities

by sqing, Apr 25, 2025, 9:19 AM

Let $a,b \in [0 ,1] .$ Prove that
$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$

This post has been edited 4 times. Last edited by sqing, Apr 25, 2025, 9:53 AM

inequalities

algebra

Geometric inequality

by ReticulatedPython, Apr 22, 2025, 5:12 PM

Let $A$ and $B$ be points on a plane such that $AB=n$ , where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$ , where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$

This post has been edited 2 times. Last edited by ReticulatedPython, Apr 22, 2025, 8:06 PM

geometry

inequalities

geometric inequality

Three variables inequality

by Headhunter, Apr 20, 2025, 6:58 AM

$\forall a\in R$ , $~\forall b\in R$ , $~\forall c \in R$
Prove that at least one of $(a-b)^{2}$ , $(b-c)^{2}$ , $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$ .

I assume that all are greater than it, but can't go more.

inequalities

algebra

proof-writing

Basic geometry

by AlexCenteno2007, Feb 9, 2025, 4:16 AM

Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE

This post has been edited 1 time. Last edited by AlexCenteno2007, Feb 9, 2025, 4:16 AM
Reason: Error

geometry

#18) Direct Pascal's Theorem

by Drunken_Master, Mar 7, 2018, 7:45 AM

Let $\omega$ and $O$ be the circumcircle and circumcenter of right triangle $ABC$ with $\angle B =90^{\circ}$ . Let $P$ be any point on the tangent to $\omega$ at $A$ other than $A$ , and suppose ray $PB$ intersects $\omega$ again at $D$ . Point $E$ lies on line $CD$ such that $AE \parallel BC$ . Prove that $P, O, E$ are collinear.
Source
Solution

This post has been edited 1 time. Last edited by pro_4_ever, Mar 8, 2018, 8:17 AM

geometry

#17) Mixtilinear Incircle and Isogonal Lines...

by Drunken_Master, Mar 2, 2018, 7:14 AM

Let $\Omega$ be the circumcircle of the triangle $ABC$ . The circle $\omega$ is tangent to the sides $AC$ and $BC$ , and it is internally tangent to the circle $\Omega$ at the point $P$ . A line parallel to $AB$ intersecting the interior of triangle $ABC$ is tangent to $\omega$ at $Q$ .

Prove that $\angle ACP = \angle QCB$

Source
Solution

This post has been edited 1 time. Last edited by pro_4_ever, Mar 2, 2018, 4:59 PM

#16) Easy Inversion!

by Drunken_Master, Mar 1, 2018, 9:56 AM

Let $\Gamma_1$ , $\Gamma_2$ , $\Gamma_3$ , $\Gamma_4$ be distinct circles such that $\Gamma_1$ , $\Gamma_3$ are externally tangent at $P$ , and $\Gamma_2$ , $\Gamma_4$ are externally tangent at the same point $P$ . Suppose that $\Gamma_1$ and $\Gamma_2$ ; $\Gamma_2$ and $\Gamma_3$ ; $\Gamma_3$ and $\Gamma_4$ ; $\Gamma_4$ and $\Gamma_1$ meet at $A$ , $B$ , $C$ , $D$ , respectively, and that all these points are different from $P$ . Prove that

$\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.$ Source
Solution

This post has been edited 1 time. Last edited by pro_4_ever, Mar 1, 2018, 10:27 AM

geometry

Inversion

Submit

Hey, what a nice blog!!!!!
by Power_Set, Jun 24, 2018, 9:53 PM
Is your blog dead ? No new post in almost 2 months.

My blog has died due to malnutrition of problems.
by integrated_JRC, May 2, 2018, 5:52 AM
@ below, I will do it soon
by pro_4_ever, Mar 1, 2018, 10:37 AM
Hi, I think we should 'hide' our solutions, as it is difficult to go up and down the blog
by Drunken_Master, Mar 1, 2018, 10:36 AM
Yeah, I created a test blog, it seems to be an AoPS problem not CSS, can't do anything about it. I'll post it in site support forum later.
by Vrangr, Feb 20, 2018, 5:48 AM
The problem is not in the CSS, I believe, as the CSS only looks after the appearance of the blog, and not the $\text{\LaTeX}$ rendering.
by WizardMath, Feb 19, 2018, 11:34 PM
I'll fix the CSS after boards of need be (I'll probably be starting my own blog too then)
by Vrangr, Feb 19, 2018, 4:41 PM
@ below,
Pls ask WizardMath...
The CSS is taken from his amazing Blog, "An Olympiad Journey"...
I know nothing abt programming.
(BTW I got permission for taking the CSS. No Copyright Issues!)
by pro_4_ever, Feb 19, 2018, 4:16 PM
Huge flaw in website design: comment gets erased if while typing $\LaTeX$ has an error and needs to be fixed.
by Vrangr, Feb 19, 2018, 12:26 PM
Upon WizardMath's Request, let us all post synthetic solutions. Well, let's declare that this blog is intended to give the reader some good problems with simple synthetic solutions. All the best. Not trying to be rude,but I might delete upcoming Bashes
by pro_4_ever, Feb 13, 2018, 4:50 PM
Here's an idea. Ban all bashes on this blog from now on. Also try posting your own previous geo solutions as storage (I hope this is allowed). This would make the quality of the blog better. Bashes are basically write-once, read-never things. Why bother?
by WizardMath, Feb 13, 2018, 4:33 PM
Somebody type Egmo 2013 Problem 1.
No bash,purely troll problem.
by QWERTYphysics, Feb 13, 2018, 4:11 PM
I hate having to see a pure geometry blog being swamped to death with bashes.

Obviously a bash is a super terrible idea. Don't do one.
Unless you are bad at synthetic.
by WizardMath, Feb 13, 2018, 2:58 PM
@ayan Done:)
by pro_4_ever, Feb 13, 2018, 1:38 PM
9th shout
by MEGAKNIGHT, Feb 13, 2018, 11:23 AM
@below Pls don't post a bash if you are thinking about posting a new entry!
by pro_4_ever, Feb 13, 2018, 11:11 AM
Wait, I thought bash is frowned upon in this blog.
by WizardMath, Feb 12, 2018, 6:57 PM
Hmmm...
Combi Geometry is accepted...
by pro_4_ever, Feb 10, 2018, 7:32 AM
Can we post Combi here?
by ayan.nmath, Feb 10, 2018, 7:31 AM
Parag dey
by Paragdey12, Feb 6, 2018, 1:17 PM
More posts please
by AnArtist, Feb 6, 2018, 2:38 AM
Second Shout!
by ccx09, Feb 3, 2018, 9:20 PM
1st shout!
by AnArtist, Jan 31, 2018, 9:20 AM