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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Wednesday at 11:40 PM by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Wednesday at 11:40 PM
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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inequality with natural parameters
jasperE3   5
N 29 minutes ago by AshAuktober
Source: S&M 2002 3&4th Grade P1
For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality
$$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$
5 replies
jasperE3
May 15, 2021
AshAuktober
29 minutes ago
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Modular Arithmetic and Integers
steven_zhang123   0
33 minutes ago
Integers \( n, a, b \in \mathbb{Z}^+ \) satisfies \( n + a + b = 30 \). If \( \alpha < b, \alpha \in \mathbb{Z^+} \), find the maximum value of $\sum_{k=1}^{\alpha} \left \lfloor \frac{kn^2 \bmod a }{b-k} \right \rfloor $.
0 replies
steven_zhang123
33 minutes ago
0 replies
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Perfect Squares and Polynomial Pursuit
steven_zhang123   0
an hour ago
Source: ZDSX 2014. ZK1 Quiz 1 Round 2 P2
Find all real number \( x \) such that \( 8x^3 - 20, 2x^5 - 2 \) are all perfect squares.
0 replies
steven_zhang123
an hour ago
0 replies
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A metrical relation for an equilateral triangle.
Virgil Nicula   1
N an hour ago by Mathzeus1024
PP19. Let an equilateral $\triangle ABC$ with the circumcircle $w$ and a point $P\in w$ so that $AB$ separates $P$ and $C$ . Denote

$\left\{\begin{array}{ccc} D\in BC & ; & PD\perp BC\\\\ E\in CA & ; & PE\perp CA\\\\ F\in AB & ; & PF\perp AB\end{array}\right\|$ . Prove that $\frac {AF^2+BD^2+CE^2}{AF\cdot CE-BD\cdot (AF+CE)}=\frac 52$ (Miguel Ochoa Sanchez, PERU).
1 reply
Virgil Nicula
Dec 15, 2014
Mathzeus1024
an hour ago
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tangent circles
george_54   5
N an hour ago by ohiorizzler1434
$ABC$ is a triangle with circumcenter $(\Omega)$ and $(\omega)$ is a circle tangent to $BC$ and internally to $(\Omega).$ The tangent
from $A$ to $(\omega)$ intersects $(\Omega)$ again at $D.$ If $T, P$ are the contact points of $(\omega)$ with $BC, AD$ respectively, prove that $CT\cdot AD=AC\cdot PD+DC\cdot PA.$
5 replies
george_54
Mar 26, 2025
ohiorizzler1434
an hour ago
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a+b+c=xyz=1 inequality in 6 variables
Marius_Avion_De_Vanatoare   8
N an hour ago by Juno_34
Source: Moldova JTST 2024 P1
Let $a,b,c,x,y,z$ be positive real numbers, such that $a+b+c=xyz=1$ Prove that:
$$ \frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1 $$When does equality hold?
8 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
Juno_34
an hour ago
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EGMO Genre Predictions
ohiorizzler1434   7
N 2 hours ago by maths_enthusiast_0001
Everybody, with EGMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
7 replies
ohiorizzler1434
Today at 4:16 AM
maths_enthusiast_0001
2 hours ago
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AM-GM FE ineq
navi_09220114   1
N 2 hours ago by InterLoop
Source: Own. Malaysian IMO TST 2025 P3
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.

Proposed by Ivan Chan Kai Chin
1 reply
navi_09220114
Mar 22, 2025
InterLoop
2 hours ago
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Classic Inequality
KHOMNYO2   1
N 3 hours ago by lbh_qys
Given positive real numbers $x,y,z$ such that $x^2 + y^2 + z^2 = 3$. Prove the following holds

$$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} + 2x + 2y + 2z \geq 9$$
1 reply
KHOMNYO2
4 hours ago
lbh_qys
3 hours ago
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Unconventional limit
RenheMiResembleRice   8
N 4 hours ago by ohiorizzler1434
Source: Shengying Mu, Kuiyang Chao
Find a function h(x) satisfying the following
8 replies
RenheMiResembleRice
5 hours ago
ohiorizzler1434
4 hours ago
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Beautiful numbers expressed as a fraction with sum of squares
MrHeccMcHecc   0
4 hours ago
An integer $k$ is called $beautiful$ if there exists $a,b,c \in \mathbb{Z}^+$ and $c \geq 2024$ such that $$k=\frac{a^2+b^2}{c^2+1}$$Determine all $beautiful$ numbers $k$.
0 replies
MrHeccMcHecc
4 hours ago
0 replies
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Unsolved NT, 3rd time posting
GreekIdiot   1
N 4 hours ago by GreekIdiot
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
1 reply
GreekIdiot
Mar 26, 2025
GreekIdiot
4 hours ago
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Unique solution
USJL   1
N 4 hours ago by pco
Source: 2025 Taiwan TST Round 2 Independent Study 2-A
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and
\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]for all $x,y\in\mathbb{R}$.

Proposed by usjl
1 reply
USJL
Yesterday at 9:00 PM
pco
4 hours ago
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Inspired by old results
sqing   1
N 4 hours ago by sqing
Source: Own
Let $ a,b,c \geq 0. $ Prove that
$$a^2+2b^2+c^2 +2abc+\frac{1}{2} \geq 2(ab+bc+ca)$$$$a^2+ \frac{1}{2} b^2+c^2 +abc+\frac{5\sqrt 5-11}{4} \geq ab+bc+ca$$$$3a^2+2b^2+3 c^2 +4abc+3\sqrt3 -\frac{9}{2}\geq4(ab+bc+ca)$$
1 reply
sqing
5 hours ago
sqing
4 hours ago
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Inspired by JK1603JK
sqing   2
N Mar 21, 2025 by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=2. $ Prove that
$$ \frac{a+b+c-3abc}{a^2b+b^2c+c^2a}\geq\frac{1}{2}$$$$ \frac{a+b+c-3abc-2}{a^2b+b^2c+c^2a}\geq\frac{1-\sqrt 6}{2}$$$$ \frac{a+b+c-3abc-1 }{a^2b+b^2c+c^2a} \geq\frac{2-\sqrt 6}{4}$$$$ \frac{a+b+c-\frac{1}{6}abc-2}{a^2b+b^2c+c^2a}\geq\frac{13}{9}-\sqrt {\frac{3}{2}}$$$$ \frac{a+b+c-abc-2}{a^2b+b^2c+c^2a}\geq\frac{7-3\sqrt 6}{6}$$
2 replies
sqing
Mar 21, 2025
sqing
Mar 21, 2025
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Inspired by JK1603JK
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Reveal topic tags
inequalities proposed
algebra
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Source: Own
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sqing
41292 posts
sqing
#1 h  Mar 21, 2025, 7:13 AM
Y by
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=2. $ Prove that
$$ \frac{a+b+c-3abc}{a^2b+b^2c+c^2a}\geq\frac{1}{2}$$$$ \frac{a+b+c-3abc-2}{a^2b+b^2c+c^2a}\geq\frac{1-\sqrt 6}{2}$$$$ \frac{a+b+c-3abc-1 }{a^2b+b^2c+c^2a} \geq\frac{2-\sqrt 6}{4}$$$$ \frac{a+b+c-\frac{1}{6}abc-2}{a^2b+b^2c+c^2a}\geq\frac{13}{9}-\sqrt {\frac{3}{2}}$$$$ \frac{a+b+c-abc-2}{a^2b+b^2c+c^2a}\geq\frac{7-3\sqrt 6}{6}$$
This post has been edited 2 times. Last edited by sqing, Mar 21, 2025, 7:26 AM
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lbh_qys
450 posts
lbh_qys
#2 h  Mar 21, 2025, 8:04 AM
Y by
sqing wrote:
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=2. $ Prove that
$$ \frac{a+b+c-3abc}{a^2b+b^2c+c^2a}\geq\frac{1}{2}$$

By AM-GM

$$(ab+bc+ca)(a+b+c) - 6abc - (a^2 b+b^2 c+c^2 a) = ab^2 + bc^2 + ca^2 -3abc \geq 0$$
Z K Y
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sqing
41292 posts
sqing
#3 h  Mar 21, 2025, 8:08 AM
Y by
Very nice.Thank bh_qys.
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