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

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

Inequalities

by sqing, Jul 12, 2024, 10:41 AM

Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$(1-abc) (1-a)(1-b)(1-c) \ge 208$ $(1+abc) (1-a)(1-b)(1-c) \le -224$ $(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840$

inequalities

algebra

Art of Problem Solving Blog

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

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

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

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

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

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

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

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

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

by sqing, Jul 12, 2024, 10:41 AM