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Inspired by old results
sqing   0
12 minutes ago
Source: Own
Let $ a , b , c>0 $and $ abc=1 $. Prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3 \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$h
0 replies
sqing
12 minutes ago
0 replies
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9 Physical or online
wimpykid   0
4 hours ago
Do you think the AoPS print books or the online books are better?

0 replies
wimpykid
4 hours ago
0 replies
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Three variables inequality
Headhunter   6
N 4 hours ago by lbh_qys
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
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.
6 replies
Headhunter
Apr 20, 2025
lbh_qys
4 hours ago
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Sequence
lgx57   8
N 5 hours ago by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
5 hours ago
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Geometric inequality
ReticulatedPython   3
N Today at 4:27 AM by ItalianZebra
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 $n.$
3 replies
ReticulatedPython
Apr 22, 2025
ItalianZebra
Today at 4:27 AM
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Transformation of a cross product when multiplied by matrix A
Math-lover1   1
N Today at 1:02 AM by greenturtle3141
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
[quote=AoPS Volume 2]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.[/quote]
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?
1 reply
Math-lover1
Yesterday at 10:29 PM
greenturtle3141
Today at 1:02 AM
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Geometry books
T.Mousavidin   4
N Today at 12:10 AM by compoly2010
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,
4 replies
T.Mousavidin
Yesterday at 4:25 PM
compoly2010
Today at 12:10 AM
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trigonometric functions
VivaanKam   3
N Yesterday at 10:08 PM by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
3 replies
VivaanKam
Yesterday at 8:29 PM
aok
Yesterday at 10:08 PM
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Inequalities
sqing   16
N Yesterday at 5:25 PM by martianrunner
Let $ a,b \in [0 ,1] . $ Prove that
$$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$$$$ \frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$$$$\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$$$$\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$$
16 replies
sqing
Apr 25, 2025
martianrunner
Yesterday at 5:25 PM
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Geometry Angle Chasing
Sid-darth-vater   6
N Yesterday at 2:18 PM by sunken rock
Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!
6 replies
Sid-darth-vater
Apr 21, 2025
sunken rock
Yesterday at 2:18 PM
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   0
Yesterday at 12:18 PM
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
0 replies
Mathgloggers
Yesterday at 12:18 PM
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max |sin x|, |sin (x+1)| > 1/3
Miquel-point   3
N Apr 15, 2025 by Mathzeus1024
Source: Romanian IMO TST 1981, Day 2 P1
Show that for every real number $x$ we have
\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]
3 replies
Miquel-point
Apr 6, 2025
Mathzeus1024
Apr 15, 2025
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max |sin x|, |sin (x+1)| > 1/3
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Source: Romanian IMO TST 1981, Day 2 P1
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Miquel-point
477 posts
Miquel-point
#1 h  Apr 6, 2025, 6:24 PM • 1 Y
Y by PikaPika999
Show that for every real number $x$ we have
\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]
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Mathzeus1024
854 posts
Mathzeus1024
#2 h  Apr 13, 2025, 2:24 PM
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The minimum of $f(x) = \max\{|\sin(x)|, |\sin(x+1)|\}$ occurs when $-\sin(x) = \sin(x+1)$ for all $x \in \mathbb{R}$, or:

$-\sin(x) = \sin(x)\cos(1)+\cos(x)\sin(1)$;

or $-\sin(x)[1+\cos(1)]=\sin(1)\sqrt{1-\sin^{2}(x)}$;

or $\sin^{2}(x)[1+\cos(1)]^2 = \sin^{2}(1)[1-\sin^{2}(x)]$;

or $\sin^{2}(x)\left[\frac{1+\cos(1)}{\sin(1)}\right]^2 = 1-\sin^{2}(x)$;

or $\left[\frac{1+\cos(1)}{\sin(1)}\right]^2 + 1=\frac{1}{\sin^{2}(x)}$;

or $|\sin(x)| = \frac{\sin(1)}{\sqrt{2[1+\cos(1)]}} = 0.47945 > \frac{1}{3}$.
This post has been edited 2 times. Last edited by Mathzeus1024, Apr 15, 2025, 10:29 AM
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jasperE3
11279 posts
jasperE3
#3 h  Apr 13, 2025, 4:05 PM
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Mathzeus1024 wrote:
$\frac{\sin(1)}{\sqrt{2[1+\cos(1)]}} = 0.47945 > \frac{1}{3}$.

how?
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Mathzeus1024
854 posts
Mathzeus1024
#4 h  Apr 15, 2025, 10:30 AM
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jasperE3 wrote:
Mathzeus1024 wrote:
$\frac{\sin(1)}{\sqrt{2[1+\cos(1)]}} = 0.47945 > \frac{1}{3}$.

how?

A quick plot of both y=|sin x| and y=|sin(x+1)| will make it apparent.
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