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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Simple inequality
sqing   7
N 4 minutes ago by sqing
Source: Daniel Sitaru
Let $a,b,c>0$ . Prove that$$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$$
7 replies
sqing
Feb 10, 2017
sqing
4 minutes ago
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Vector Vortex
steven_zhang123   1
N 4 minutes ago by Mathzeus1024
Source: NS Issue 1 P3 (2014.4)
Let $v_{1}, v_{2}, \cdots, v_{n}$ be $n$ unit vectors on a plane, where $n$ is an odd number. Prove that there exist $\varepsilon _i\in \left \{ -1,1 \right \} $ for $i=1,2,\cdots,n$ such that $\left | \sum_{i=1}^{n} \varepsilon_i v_i \right | \le 1.$
1 reply
steven_zhang123
Feb 15, 2025
Mathzeus1024
4 minutes ago
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China Northern MO 2009 p4 CNMO
parkjungmin   0
4 minutes ago
Source: China Northern MO 2009 p4 CNMO P4
The problem is too difficult.
0 replies
parkjungmin
4 minutes ago
0 replies
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The Appetizer of Iran NT2023
alinazarboland   6
N 18 minutes ago by A22-
Source: Iran MO 3rd round 2023 NT exam , P1
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
6 replies
alinazarboland
Aug 17, 2023
A22-
18 minutes ago
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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 5 hours ago 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
5 hours ago
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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
J
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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
0 replies
J
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N.S. condition of passing a fixed point for a function
Kunihiko_Chikaya   1
N Yesterday at 11:29 AM by Mathzeus1024
Let $ f(t)$ be a function defined in any real numbers $ t$ with $ f(0)\neq 0.$ Prove that on the $ x-y$ plane, the line $ l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $ y$ axis in regardless of the value of $ t$ if only if $ f(t)$ is a linear function in $ t$.
1 reply
Kunihiko_Chikaya
Sep 6, 2009
Mathzeus1024
Yesterday at 11:29 AM
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Inspired by Omerking
sqing   2
N Apr 16, 2025 by sqing
Source: Own
Let $ a,b,c>0 $ and $ ab+bc+ca\geq \dfrac{1}{3}. $ Prove that
$$ ka+ b+kc\geq \sqrt{\frac{4k-1}{3}}$$Where $ k\geq 1.$$$ 4a+ b+4c\geq \sqrt{5}$$
2 replies
sqing
Apr 16, 2025
sqing
Apr 16, 2025
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Inspired by Omerking
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Reveal topic tags
inequalities proposed
algebra
inequalities
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G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41897 posts
sqing
#1 h  Apr 16, 2025, 5:11 AM
Y by
Let $ a,b,c>0 $ and $ ab+bc+ca\geq \dfrac{1}{3}. $ Prove that
$$ ka+ b+kc\geq \sqrt{\frac{4k-1}{3}}$$Where $ k\geq 1.$$$ 4a+ b+4c\geq \sqrt{5}$$
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 5:16 AM
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lbh_qys
557 posts
lbh_qys
#2 h  Apr 16, 2025, 5:40 AM • 1 Y
Y by kiyoras_2001
sqing wrote:
Let $ a,b,c>0 $ and $ ab+bc+ca\geq \dfrac{1}{3}. $ Prove that
$$ ka+ b+kc\geq \sqrt{\frac{4k-1}{3}}$$Where $ k\geq 1.$$$ 4a+ b+4c\geq \sqrt{5}$$

\[ 4ka + 4b + 4kc = (4k-1)(a+c) + (a+c+4b) \geq 2\sqrt{(4k-1)(a+c)(a+c+4b)} \geq 2\sqrt{(4k-1)\cdot 4(ab+bc+ca)} \geq 4\sqrt{\frac{4k-1}{3}}. \]
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sqing
41897 posts
sqing
#3 h  Apr 16, 2025, 12:28 PM
Y by
Very very nice.Thank lbh_qys.
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