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

Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
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
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
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
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
J
H
k i Suggestion Form
jwelsh   0
May 6, 2021
Hello!

Given the number of suggestions we’ve been receiving, we’re transitioning to a suggestion form. If you have a suggestion for the AoPS website, please submit the Google Form:
Suggestion Form

To keep all new suggestions together, any new suggestion threads posted will be deleted.

Please remember that if you find a bug outside of FTW! (after refreshing to make sure it’s not a glitch), make sure you’re following the How to write a bug report instructions and using the proper format to report the bug.

Please check the FTW! thread for bugs and post any new ones in the For the Win! and Other Games Support Forum.
0 replies
jwelsh
May 6, 2021
0 replies
J
H
k i Read me first / How to write a bug report
slester   3
N May 4, 2019 by LauraZed
Greetings, AoPS users!

If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.

Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.

For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.

What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.

If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.

The subject
The subject line should explain as clearly as possible what went wrong.

Bad: Forum doesn't work
Good: Switching between threads quickly shows blank page.

The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.

Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:

If your computer or tablet is school issued, please indicate this under Additional information.

Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.

How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.

Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.

Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.

In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.

It'll look something like this:
IMAGE
3 replies
slester
Apr 9, 2015
LauraZed
May 4, 2019
J
H
k i Community Safety
dcouchman   0
Jan 18, 2018
If you find content on the AoPS Community that makes you concerned for a user's health or safety, please alert AoPS Administrators using the report button (Z) or by emailing sheriff@aops.com . You should provide a description of the content and a link in your message. If it's an emergency, call 911 or whatever the local emergency services are in your country.

Please also use those steps to alert us if bullying behavior is being directed at you or another user. Content that is "unlawful, harmful, threatening, abusive, harassing, tortuous, defamatory, vulgar, obscene, libelous, invasive of another's privacy, hateful, or racially, ethnically or otherwise objectionable" (AoPS Terms of Service 5.d) or that otherwise bullies people is not tolerated on AoPS, and accounts that post such content may be terminated or suspended.
0 replies
dcouchman
Jan 18, 2018
0 replies
J
H
Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   59
N 6 minutes ago by math-olympiad-clown
Source: BMO 2023 Problem 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
Proposed by Nikola Velov, Macedonia
59 replies
cretanman
May 10, 2023
math-olympiad-clown
6 minutes ago
J
H
Inspired by Adhyayan Jana
sqing   2
N 21 minutes ago by sqing
Source: Own
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 +c^2 = b^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$Let $a,b,c,d>0,a^2 + d^2-ad = b^2 + c^2 + bc $ aand $ a^2 +c^2 = b^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{\sqrt 3}{2}$$Let $a,b,c,d>0,a^2 + d^2 - ad = b^2 + c^2 + bc $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that $$ \frac{ab+cd}{ad+bc} =\frac{\sqrt 3}{2}$$
2 replies
+1 w
sqing
2 hours ago
sqing
21 minutes ago
J
H
Concurrent lines
syk0526   28
N 33 minutes ago by alexanderchew
Source: North Korea Team Selection Test 2013 #1
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
28 replies
syk0526
May 17, 2014
alexanderchew
33 minutes ago
J
H
Equal angles (a very old problem)
April   56
N 34 minutes ago by Ilikeminecraft
Source: ISL 2007, G3, VAIMO 2008, P5
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD = \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP = \angle DAQ$.

Author: Vyacheslav Yasinskiy, Ukraine
56 replies
1 viewing
April
Jul 13, 2008
Ilikeminecraft
34 minutes ago
J
H
k Who was the FIRST AoPS user (its not cisco man)
EaZ_Shadow   8
N Yesterday at 2:11 AM by Craftybutterfly
Alright, if you look at any other user's page, you'll see a number attached to artofproblemsolving.com/community/users/(number). That number signifies the place in when that person entered. Alright. If i got the number 1163, I was the 1163rd person to join. But then who was the first one? With the number 1?
8 replies
EaZ_Shadow
May 25, 2025
Craftybutterfly
Yesterday at 2:11 AM
J
H
k Request for the access of private marathons
Vulch   4
N Monday at 2:32 PM by bpan2021
To all AoPS users and admin,
Sometimes I came across the marathon(i .e number theory marathon, functional equation marathon etc) which allow access only after submitting log in request.There is no other way to access the question related to that marathons.It would be glad to open all private marathons publicly.Thank you!
4 replies
Vulch
May 25, 2025
bpan2021
Monday at 2:32 PM
J
H
k Mathcounts trainer slow?
HM2018   0
May 26, 2025
Summary of problem: The Mathcounts Trainer is slow again.
URL Page: https://artofproblemsolving.com/mathcounts_trainer/play
If you answer a question, it takes so long for it to show you the answer.
Behavior (expected): Should show the answer pretty quickly (wasn't Mathcounts Trainer updated recently?)
0 replies
HM2018
May 26, 2025
0 replies
J
H
Character encoding?
char0221   0
May 26, 2025
Summary of the problem: When using the "js", "java", "c", and "cpp" code tags, a left square bracket (i.e. "[") does not show. Instead, "[" appears.
Page URL: Any AoPS message board or community area.
Steps to reproduce:
1. Create a new message.
2. Inside, use a coding language template. I have not tried the following: Ruby, Go, C# (is there even on for C#?), and Python
3. Type "[".
4. Watch in amazement as your code is ruined.
Expected behavior: Inside the template, should make "[" appear.
Frequency: 100%
Operating system(s): macOS Sequoia
Browser(s), including version: Safari 18.4
Additional information: See below.
-|-leftbracket-|-]

It is "[" next to "]". Only the left bracket does not render.
0 replies
char0221
May 26, 2025
0 replies
J
H
k Who is Halp! ? (resolvedd)
A7456321   8
N May 25, 2025 by JohannIsBach
Is Halp! a bot? This user has been posting questions in nearly all of my AoPS classes when the user isn't a part of the class, and this user has 150k posts.
8 replies
A7456321
May 24, 2025
JohannIsBach
May 25, 2025
J
H
k How to create a poll?
whwlqkd   32
N May 25, 2025 by Yiyj
How to create a poll in aops?
32 replies
whwlqkd
May 24, 2025
Yiyj
May 25, 2025
J
H
k banned myself from own blog
Spacepandamath13   8
N May 25, 2025 by sultanine
I got curious and decided to see if I can ban myself from my own blog.
can site admins give it back? it says site admins are the administrators of this blog

I honestly don't know where I come up with stuff like this
8 replies
Spacepandamath13
May 24, 2025
sultanine
May 25, 2025
J
H
k resolved!
JohannIsBach   6
N May 24, 2025 by bpan2021
hi srry if this is in the rong place i didnt no where 2 put it i was wondering how u find a user? i tried using the search but they dont have any posts? dont no wat 2 do...
6 replies
JohannIsBach
May 24, 2025
bpan2021
May 24, 2025
J
H
k Introducing myself at AoPS, and what's your magic wand?
asuth_asuth   1193
N May 23, 2025 by Penguin117
Hi!

I'm Andrew Sutherland. I'm the new Chief Product Officer at AoPS. As you may have read, Richard is retiring and Ben Kornell and I are working together to lead the company now. I'm leading all the software and digital stuff at AoPS. I just wanted to say hello and introduce myself! I'm really excited to be part of the special community that is AoPS.

Previously, I founded Quizlet as a 15-year-old high school student. I did Course 6 at MIT and then left to lead Quizlet full-time for a total of 14 years. I took a few years off and now I'm doing AoPS! I wrote more about all that on my blog: https://asuth.com/im-joining-aops

I have a question for all of you. If you could wave a magic wand, and change anything about AoPS, what would it be? All suggestions welcome! Thank you.
1193 replies
asuth_asuth
Mar 30, 2025
Penguin117
May 23, 2025
J
H
k how 2 play reaper?
JohannIsBach   3
N May 22, 2025 by JohannIsBach
hi srry if this is in the rong place i dont no where else 2 put it how do u play reaper? and is htere a link 2 the game? just wondering
3 replies
JohannIsBach
May 22, 2025
JohannIsBach
May 22, 2025
J
H
J
H
Number Theory
AnhQuang_67   3
N Apr 24, 2025 by GreekIdiot
Source: HSGSO 2024
Let $p$ be an odd prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
3 replies
AnhQuang_67
Apr 24, 2025
GreekIdiot
Apr 24, 2025
J
Number Theory
G H J
Reveal topic tags
number theory
prime numbers
L
G H BBookmark kLocked kLocked NReply
Source: HSGSO 2024
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
AnhQuang_67
57 posts
AnhQuang_67
#1 h  Apr 24, 2025, 1:44 PM
Y by
Let $p$ be an odd prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
This post has been edited 1 time. Last edited by AnhQuang_67, Apr 24, 2025, 2:54 PM
Reason: sorry, my bad
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
269 posts
GreekIdiot
#2 h  Apr 24, 2025, 2:26 PM
Y by
$p$ is an even prime? Then $p=2$ thus $a_n=0,1 \: mod \: 2$
and since $a_1,a_2$ are not of the same parity there doesnt exist such $k$
I assume you mean 'odd' prime
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
AnhQuang_67
57 posts
AnhQuang_67
#3 h  Apr 24, 2025, 2:55 PM
Y by
GreekIdiot wrote:
$p$ is an even prime? Then $p=2$ thus $a_n=0,1 \: mod \: 2$
and since $a_1,a_2$ are not of the same parity there doesnt exist such $k$
I assume you mean 'odd' prime
Oops, sorry about that. It is "odd"
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
269 posts
GreekIdiot
#4 h  Apr 24, 2025, 3:14 PM
Y by
Maybe Helpful
Z K Y
N Quick Reply
G
H
=
a