Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

We have your learning goals covered with Spring and Summer courses available. Enroll today!
Site Support Tech support and questions about AoPS classes and materials
Tech support and questions about AoPS classes and materials
3 M G
BBookmark  VNew Topic kLocked
Site Support Tech support and questions about AoPS classes and materials
Tech support and questions about AoPS classes and materials
3 M G
BBookmark  VNew Topic kLocked
t 
18,559 topics
w
4 users
TOPICS
No tags match your search
M
Alcumus
blogs
LaTeX
search
FTW
geometry
Support
email
MATHCOUNTS
function
AMC
AoPSwiki
Bug
Reaper
Forums
AIME
suggestion
poll
site support
AMC 10
Asymptote
AoPS
AoPS classes
videos
posts
Glitch
calculus
pms
Suggestions
probability
help
classroom
number theory
classes
AMC 8
AoPS Books
question
Forum
glitch or bug
Greed Control
3D geometry
AIME I
USA(J)MO
Tags
Homework
Avatars
Friends
BBCode
upvotes
Post
No tags match your search
M
G
Topic
First Poster
Last Poster
H
a My Retirement & New Leadership at AoPS
rrusczyk   1334
N 3 minutes ago by poodle
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
1334 replies
rrusczyk
Monday at 6:37 PM
poodle
3 minutes ago
J
H
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.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21


Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Sunday, Mar 16 - Aug 24
Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Mar 2, 2025
0 replies
J
H
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
J
H
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
J
H
AoPS wiki loading slowly?
EaZ_Shadow   12
N 20 minutes ago by Craftybutterfly
I don't know why, but why is that when I try loading to AoPS Wiki, it loads really slow? (I'm using an iPad)
12 replies
EaZ_Shadow
Yesterday at 9:18 PM
Craftybutterfly
20 minutes ago
J
H
Flee Jumping on Number Line
utkarshgupta   23
N an hour 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
an hour ago
J
H
Smallest value of |253^m - 40^n|
MS_Kekas   3
N an hour 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
MS_Kekas
Jan 28, 2024
imagien_bad
an hour ago
J
H
Operating on lamps in a circle
anantmudgal09   7
N an hour 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
2 viewing
anantmudgal09
Dec 9, 2017
hectorleo123
an hour ago
J
H
2025 Caucasus MO Seniors P1
BR1F1SZ   3
N an hour 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
4 hours ago
Mathdreams
an hour ago
J
H
IMO 2018 Problem 2
juckter   95
N an hour 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
an hour ago
J
H
Long condition for the beginning
wassupevery1   2
N an hour 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
an hour ago
J
H
Inspired by IMO 1984
sqing   0
an hour 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
an hour ago
0 replies
J
H
Prime-related integers [CMO 2018 - P3]
Amir Hossein   15
N 2 hours ago by Ilikeminecraft
Source: 2018 Canadian Mathematical Olympiad - P3
Two positive integers $a$ and $b$ are prime-related if $a = pb$ or $b = pa$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.

Note that $1$ and $n$ are included as divisors.
15 replies
Amir Hossein
Mar 31, 2018
Ilikeminecraft
2 hours ago
J
H
Inspired by IMO 1984
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +17abc\leq\frac{8000}{7803}$$$$a^2+b^2+ ab +\frac{163}{10}abc\leq\frac{7189057}{7173630}$$$$a^2+b^2+ ab +16.23442238abc\le1$$
2 replies
sqing
Yesterday at 3:04 PM
sqing
2 hours ago
J
H
2025 Caucasus MO Juniors P6
BR1F1SZ   1
N 2 hours ago by maromex
Source: Caucasus MO
A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$
1 reply
BR1F1SZ
3 hours ago
maromex
2 hours ago
J
H
k Staff, Please confirm or deny this conjecture
Mango8000   1
N Yesterday at 4:37 PM by jlacosta
It’s seems that there are suspicions about AoPS selling Beast Academy to another company. Is that true? Becuase AoPS online and Beast Academy are connected and it will affect us. I hope that AoPS decides to keep it, but if not, there really isn’t anything we can do.
1 reply
Mango8000
Monday at 9:13 PM
jlacosta
Yesterday at 4:37 PM
J
H
k spotted in blogroll
Major_Monogram   8
N Yesterday at 1:06 PM by Embershed97
I saw this on the AoPS Blogroll. closing it, the page worked normally.
8 replies
Major_Monogram
Mar 22, 2025
Embershed97
Yesterday at 1:06 PM
J
H
Office Hours not starting yet?
GAMER100   2
N Yesterday at 12:08 AM by k1glaucus
Several posts have been made about this years before, but the office hours banner hasn't appeared yet even though I refreshed the page multiple times. It has been 50 minutes since when office hours should have started and no mods have responded to a question I posted 3 hours ago (relative to last edit). Can others confirm?
2 replies
GAMER100
Monday at 11:51 PM
k1glaucus
Yesterday at 12:08 AM
J
H
J
H
k How to add members to forums?
Bnn81351   2
N Mar 20, 2025 by Craftybutterfly
How do I add members to forums?
2 replies
Bnn81351
Mar 20, 2025
Craftybutterfly
Mar 20, 2025
J
How to add members to forums?
G H J
Reveal topic tags
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Bnn81351
43 posts
Bnn81351
#1 h  Mar 20, 2025, 12:55 AM
Y by
How do I add members to forums?
Z Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
maxamc
471 posts
maxamc
#2 h  Mar 20, 2025, 1:00 AM
Y by
press add members after press the i button
This post has been edited 1 time. Last edited by maxamc, Mar 20, 2025, 1:00 AM
Z Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
249 posts
Craftybutterfly
#3 h  Mar 20, 2025, 5:22 AM
Y by
If public, share forum link. If not, press the i next to the forum name and go to the permissions.
Attachments:
This post has been edited 1 time. Last edited by Craftybutterfly, Mar 20, 2025, 5:29 AM
Reason: iphone
Z Y
G
H
=
a