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a My Retirement & New Leadership at AoPS
rrusczyk   1345
N 29 minutes ago by GoodGamer123
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
1345 replies
rrusczyk
Monday at 6:37 PM
GoodGamer123
29 minutes ago
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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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You see?
Spacepandamath13   15
N 4 minutes ago by Craftybutterfly
Why does $1+1=3$?

You see, if I give CaseOh $1$ apple, and you give him $1$ apple, he will throw up $1$ apple that he had previously eaten, leaving their $3$ apples in the pile.
15 replies
Spacepandamath13
6 hours ago
Craftybutterfly
4 minutes ago
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The daily problem!
Leeoz   22
N 21 minutes ago by HacheB2031
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]
22 replies
Leeoz
Mar 21, 2025
HacheB2031
21 minutes ago
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9 Am I going insane?
PenguFish   9
N 32 minutes ago by Skillfulcotton
I feel stupid honestly, and fyi, I did go over every question. Focus is counting with symmetry
9 replies
PenguFish
4 hours ago
Skillfulcotton
32 minutes ago
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Additive Combinatorics!
EthanWYX2009   3
N an hour ago by flower417477
Source: 2025 TST 15
Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\]\[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\]\[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\]Show that \( |A|^2 \leq |B| \cdot |C| \).
3 replies
EthanWYX2009
Yesterday at 12:49 AM
flower417477
an hour ago
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Inspired by IMO 1984
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that
$$a+b+\frac{9}{5}c\geq\frac{9}{8}$$$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+ ab +18abc\geq\frac{343}{324} $. Prove that
$$a+b+\frac{6}{5}c\geq\frac{7\sqrt 7}{18}$$$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$
0 replies
sqing
2 hours ago
0 replies
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equal angles
jhz   2
N 2 hours ago by YaoAOPS
Source: 2025 CTST P16
In convex quadrilateral $ABCD, AB \perp AD, AD = DC$. Let $ E$ be a point on side $BC$, and $F$ be a point on the extension of $DE$ such that $\angle ABF = \angle DEC>90^{\circ}$. Let $O$ be the circumcenter of $\triangle CDE$, and $P$ be a point on the side extension of $FO$ satisfying $FB =FP$. Line BP intersects AC at point Q. Prove that $\angle AQB =\angle DPF.$
2 replies
jhz
5 hours ago
YaoAOPS
2 hours ago
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Flee Jumping on Number Line
utkarshgupta   23
N 2 hours ago by Ilikeminecraft
Source: All Russian Olympiad 2015 11.5
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
23 replies
utkarshgupta
Dec 11, 2015
Ilikeminecraft
2 hours ago
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Smallest value of |253^m - 40^n|
MS_Kekas   3
N 2 hours ago by imagien_bad
Source: Kyiv City MO 2024 Round 1, Problem 9.5
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.

Proposed by Oleksii Masalitin
3 replies
1 viewing
MS_Kekas
Jan 28, 2024
imagien_bad
2 hours ago
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Operating on lamps in a circle
anantmudgal09   7
N 2 hours ago by hectorleo123
Source: India Practice TST 2017 D2 P3
There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either on or off. Every second, each lamp undergoes a change according to the following rule:

(a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is off right now. (Indices taken mod $n$.)

(b) Otherwise, $L_i$ is on right now.

Initially, all the lamps are off, except for $L_1$ which is on. Prove that for infinitely many integers $n$ all the lamps will be off eventually, after a finite amount of time.
7 replies
anantmudgal09
Dec 9, 2017
hectorleo123
2 hours ago
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2025 Caucasus MO Seniors P1
BR1F1SZ   3
N 2 hours ago by Mathdreams
Source: Caucasus MO
For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$[list=a]
[*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation.
[*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation.
[/list]
(Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)
3 replies
BR1F1SZ
5 hours ago
Mathdreams
2 hours ago
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IMO 2018 Problem 2
juckter   95
N 2 hours ago by Marcus_Zhang
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
95 replies
juckter
Jul 9, 2018
Marcus_Zhang
2 hours ago
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Long condition for the beginning
wassupevery1   2
N 3 hours ago by wassupevery1
Source: 2025 Vietnam IMO TST - Problem 1
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$holds for all positive rational numbers $x, y$.
2 replies
wassupevery1
Yesterday at 1:49 PM
wassupevery1
3 hours ago
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Inspired by IMO 1984
sqing   0
3 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +24abc\leq\frac{81}{64}$$Equality holds when $a=b=\frac{3}{8},c=\frac{1}{4}.$
$$a^2+b^2+ ab +18abc\leq\frac{343}{324}$$Equality holds when $a=b=\frac{7}{18},c=\frac{2}{9}.$
0 replies
sqing
3 hours ago
0 replies
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Newton Sums
mithu542   4
N 3 hours ago by aidan0626
Should I memorize Newton's Sums? I already know the first two, but do you think it is worthy to memorize some more (especially 3)?

Website about Newton Sums
4 replies
mithu542
4 hours ago
aidan0626
3 hours ago
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1/a+1/b+1/c=6/7 - some fun
236factorial   7
N Feb 17, 2006 by pascal_1623
If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{6}{7}$, where a, b, and c are positive integers, what is the smallest value of a+b+c?
7 replies
236factorial
Feb 11, 2006
pascal_1623
Feb 17, 2006
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1/a+1/b+1/c=6/7 - some fun
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236factorial
4555 posts
236factorial
#1 h  Feb 11, 2006, 2:25 AM • 2 Y
Y by Adventure10, Mango247
If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{6}{7}$, where a, b, and c are positive integers, what is the smallest value of a+b+c?
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robinhe
1352 posts
robinhe
#2 h  Feb 11, 2006, 7:22 PM • 3 Y
Y by Adventure10, Mango247, Mango247
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X_wing1redleader
106 posts
X_wing1redleader
#3 h  Feb 11, 2006, 10:19 PM • 2 Y
Y by Adventure10, Mango247
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redsmartie
85 posts
redsmartie
#4 h  Feb 11, 2006, 10:54 PM • 2 Y
Y by Adventure10, Mango247
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b-flat
1903 posts
b-flat
#5 h  Feb 14, 2006, 12:12 AM • 2 Y
Y by Adventure10, Mango247
redsmartie wrote:
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Here is your answer written in $\LaTeX$ so it is easier to read (I hope I didn't make any mistakes :D ):

Click to reveal hidden text
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moogra
1330 posts
moogra
#6 h  Feb 16, 2006, 2:36 AM • 2 Y
Y by Adventure10, Mango247
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236factorial
4555 posts
236factorial
#7 h  Feb 16, 2006, 3:24 AM • 2 Y
Y by Adventure10, Mango247
moogra wrote:
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It's actually not that sad, because this problem actually requires some luck. A systematic way of doing this problem would take a while and is probably too advanced for a chapter level problem.
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pascal_1623
907 posts
pascal_1623
#8 h  Feb 17, 2006, 2:15 AM • 2 Y
Y by Adventure10, Mango247
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