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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 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
Mar 26, 2025
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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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Great orz
Hip1zzzil   7
N 5 minutes ago by sansae
Source: FKMO 2025 P5
$S={1,2,...,1000}$ and $T'=\left\{ 1001-t|t \in T\right\}$.
A set $P$ satisfies the following three conditions:
$1.$ All elements of $P$ are a subset of $S$.
$2. A,B \in P \Rightarrow A \cap B \neq \O$
$3. A \in P \Rightarrow A' \in P$
Find the maximum of $|P|$.
7 replies
Hip1zzzil
Today at 4:54 AM
sansae
5 minutes ago
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IMO ShortList 2001, combinatorics problem 3
orl   34
N 9 minutes ago by Marcus_Zhang
Source: IMO ShortList 2001, combinatorics problem 3, HK 2009 TST 2 Q.2
Define a $ k$-clique to be a set of $ k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.
34 replies
orl
Sep 30, 2004
Marcus_Zhang
9 minutes ago
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Long polynomial factorization
wassupevery1   7
N 12 minutes ago by giangtruong13
Source: 2025 Vietnam IMO TST - Problem 6
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$$Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
7 replies
wassupevery1
Mar 26, 2025
giangtruong13
12 minutes ago
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orz otl fr
Hip1zzzil   7
N 23 minutes ago by sansae
Source: FKMO 2025 P3
An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$.
$I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively.
Prove $l_{1},l_{2},l_{3}$ meet at one point.
7 replies
Hip1zzzil
Yesterday at 10:23 AM
sansae
23 minutes ago
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Diophantine No Sols
anantmudgal09   7
N 25 minutes ago by giangtruong13
Source: The 1st India-Iran Friendly Competition Problem 4
Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$
Proposed by Navid Safaei
7 replies
anantmudgal09
Jun 13, 2024
giangtruong13
25 minutes ago
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Functional Equations Marathon March 2025
Levieee   26
N 28 minutes ago by Sedro
1. before posting another problem please try your best to provide the solution to the previous solution because we don't want a backlog of many problems
2.one is welcome to send functional equations involving calculus (mainly basic real analysis type of proofs) as long it is of the form $\text{"find all functions:"}$
26 replies
Levieee
Mar 17, 2025
Sedro
28 minutes ago
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Very easy inequality
Bugi   32
N 30 minutes ago by Alex-131
Source: JBMO Shortlist 2002
Let $ a,b,c$ be positive real numbers. Prove the inequality:
$ \frac {a^3}{b^2} + \frac {b^3}{c^2} + \frac {c^3}{a^2}\ge \frac {a^2}{b} + \frac {b^2}{c} + \frac {c^2}{a}$
32 replies
1 viewing
Bugi
Nov 12, 2008
Alex-131
30 minutes ago
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FE f(x)f(y)+1=f(x+y)+f(xy)+xy(x+y-2)
steven_zhang123   5
N 31 minutes ago by pco
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, we have $f(x)f(y)+1=f(x+y)+f(xy)+xy(x+y-2)$.
5 replies
steven_zhang123
Yesterday at 11:27 PM
pco
31 minutes ago
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polynomial
hanzo.ei   0
35 minutes ago
Source: from a book
Given a polynomial \( P(x) \) satisfying \( P(0) = 0 \), \( P(1) = 1 \), and for \( n \) (\( n \geq 2, n \in \mathbb{N} \)) positive real numbers \( k_1, k_2, \dots, k_n \). Prove that there exists a strictly increasing sequence of real numbers \( (a_i)_{i=1}^{n} \subset (0,1) \) such that
\[ \sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i. \]
0 replies
hanzo.ei
35 minutes ago
0 replies
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Inspired by old results
sqing   6
N 41 minutes ago by MathsII-enjoy
Source: Own
Let $ a,b,c > 0 $ and $ a+b+c +abc =4. $ Prove that
$$ a^2 + b^2 + c^2 + 3 \geq 2( ab+bc + ca )$$Let $ a,b,c > 0 $ and $ ab+bc+ca+abc=4. $ Prove that
$$ a^2 + b^2 + c^2 + 2abc \geq 5$$
6 replies
sqing
Mar 27, 2025
MathsII-enjoy
41 minutes ago
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$$ac=bd$$
sqing   2
N 43 minutes ago by Acorn-SJ
Source: Own
Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ abcd\ge 9.$ Prove that$$ac=bd$$Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ ad+bc \ge 6.$ Prove that$$ac=bd$$Let $ a,b,c,d $ be reals such that $ a^2+b^2=4,c^2+d^2=9 $ and $ab+cd \geq \frac{13}{2}.$ Prove that$$ac=bd$$




2 replies
sqing
an hour ago
Acorn-SJ
43 minutes ago
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Heavy config geo involving mixtilinear
Assassino9931   1
N 44 minutes ago by VicKmath7
Source: Bulgaria Spring Mathematical Competition 2025 12.4
Let $ABC$ be an acute-angled triangle \( ABC \) with \( AC > BC \) and incenter \( I \). Let \( \omega \) be the mixtilinear circle at vertex \( C \), i.e. the circle internally tangent to the circumcircle of \( \triangle ABC \) and also tangent to lines \( AC \) and \( BC \). A circle \( \Gamma \) passes through points \( A \) and \( B \) and is tangent to \( \omega \) at point \( T \), with \( C \notin \Gamma \) and \( I \) being inside \( \triangle ATB \). Prove that:
$$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$
1 reply
Assassino9931
2 hours ago
VicKmath7
44 minutes ago
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Product of cosines subject to product of sines
Assassino9931   1
N an hour ago by RagvaloD
Source: Bulgaria Spring Mathematical Competition 2025 11.2
Let $\alpha, \beta$ be real numbers such that $\sin\alpha\sin\beta=\frac{1}{3}$. Prove that the set of possible values of $\cos \alpha \cos \beta$ is the interval $\left[-\frac{2}{3}, \frac{2}{3}\right]$.
1 reply
Assassino9931
3 hours ago
RagvaloD
an hour ago
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A colouring game on a grid
Tintarn   2
N an hour ago by math-olympiad-clown
Source: Baltic Way 2024, Problem 8
Let $a$, $b$, $n$ be positive integers such that $a + b \leq n^2$. Alice and Bob play a game on an (initially uncoloured) $n\times n$ grid as follows:
- First, Alice paints $a$ cells green.
- Then, Bob paints $b$ other (i.e.uncoloured) cells blue.
Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of $a$, $b$ and $n$, who has a winning strategy.
2 replies
Tintarn
Nov 16, 2024
math-olympiad-clown
an hour ago
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cyclic ineq not tight
RainbowNeos   1
N Mar 27, 2025 by lbh_qys
Source: own
Given $n\geq 3$ and $x_i\geq 0, 1\leq i\leq n$ with sum $1$. Show that
\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]where $x_{n+1}=x_1$.
1 reply
RainbowNeos
Mar 26, 2025
lbh_qys
Mar 27, 2025
J
cyclic ineq not tight
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Reveal topic tags
inequalities
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Source: own
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RainbowNeos
195 posts
RainbowNeos
#1 h  Mar 26, 2025, 2:24 PM • 1 Y
Y by sky.mty
Given $n\geq 3$ and $x_i\geq 0, 1\leq i\leq n$ with sum $1$. Show that
\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]where $x_{n+1}=x_1$.
This post has been edited 1 time. Last edited by RainbowNeos, Mar 26, 2025, 2:25 PM
Reason: typo
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The post below has been deleted. Click to close.
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lbh_qys
450 posts
lbh_qys
#2 h  Mar 27, 2025, 4:24 AM • 3 Y
Y by Filipjack, RainbowNeos, kiyoras_2001
Since the minimum value is not greater than the mean value, we have
$$\min\{x_i^2, x_{i+1}\} = \min\{x_i^2, x_i^2, x_{i+1}\} \leq \operatorname{GM}(x_i^2, x_i^2, x_{i+1}) = \sqrt[3]{x_i^2 \cdot x_i^2 \cdot x_{i+1}}.$$
Hence, by AM-GM inequality,
$$\sum \min\{x_i^2, x_{i+1}\} \leq \sum \sqrt[3]{x_i^4 x_{i+1}}\leq \sum \frac{\frac{1}{2} x_i + x_i^2 + 2 x_i x_{i+1} }{3} \leq \frac{1}{6} \sum x_i + \frac{1}{3}\left(\sum x_i \right)^2 = \frac{1}{2}$$
This post has been edited 2 times. Last edited by lbh_qys, Mar 27, 2025, 4:28 AM
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