School starts soon! Add problem solving to your schedule with our math, science, and/or contest classes!

G
Topic
First Poster
Last Poster
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
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
All even perfect numbers >6 equal x^3 + y^3 + z^3
TUAN2k8   4
N 2 minutes ago by Lufin
Source: 2025 VIASM summer challenge P5
Let $n$ be an even positive integer greater than 6 such that $2n$ is equal to the sum of all distinct positive divisors of $n$. Prove that there exist distinct integers $x, y, z$ satisfying:
\[
x^3 + y^3 + z^3 = n.
\]
4 replies
1 viewing
TUAN2k8
Jul 28, 2025
Lufin
2 minutes ago
Sword in row
Eeightqx   0
4 minutes ago
Source: 2025 China South East Mathematical Olympiad Grade10 P7
We put some swords which can only face to up, down, left and right into a $n\times n$ grid with only 1 sword in a row and 1 sword in a column to form a "$n$ star sword in row". A square is said controlled if a sword is put in it or it is pointed by some swords. Try to work out the maximum number of the squares which is controlled in a $n$ star sword in row and when it gets its maximum, there are how many situations.

Below is an example of a $4$ star sword in row, swords are represented by arrows and the controlled squares are colored grey. Here 8 squares are controlled.
0 replies
Eeightqx
4 minutes ago
0 replies
Payable numbers
mathscrazy   15
N 14 minutes ago by Cats_on_a_computer
Source: INMO 2025/6
Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.

Proposed by Shantanu Nene
15 replies
mathscrazy
Jan 19, 2025
Cats_on_a_computer
14 minutes ago
Root of polynomial
Eeightqx   5
N 15 minutes ago by MathsII-enjoy
Source: 2025 China South East Mathematical Olympiad Grade10 P5
Suppose $\alpha,\,\beta,\,\gamma\ne1$ are roots of $f(x)=x^3+ax^2+bx+a+2\in\mathbb R[x]$ where $2a+b\ge24$. Prove that
$$\dfrac1{\sqrt[3]{\alpha-1}}+\dfrac1{\sqrt[3]{\beta-1}}+\dfrac1{\sqrt[3]{\gamma-1}}\le0.$$
5 replies
Eeightqx
an hour ago
MathsII-enjoy
15 minutes ago
k Deleting forums
mbt96   2
N Yesterday at 4:08 PM by PikaPika999
How do you delete forums?
2 replies
mbt96
Yesterday at 4:02 PM
PikaPika999
Yesterday at 4:08 PM
AoPS randomly capitalizing tags
ajovanovic   8
N Yesterday at 4:07 PM by PikaPika999
Sometimes, when I create a new post on my forum and add a tag, the tag's first letter automatically capitalizes the first letter! What is causing this, and is there a way to stop it?
8 replies
ajovanovic
Tuesday at 10:37 PM
PikaPika999
Yesterday at 4:07 PM
k Security Problem
whwlqkd   4
N Yesterday at 3:10 PM by whwlqkd
If other person is using my phone, he often makes something in my phone bad(like sending bad messages). How to enhance my phone’s security? I think the reason why he often makes something in my phone bad is my weak security(I do not even do 2nd certification.)
4 replies
whwlqkd
Yesterday at 10:18 AM
whwlqkd
Yesterday at 3:10 PM
k New user cannot post (with latex?)
principle9   2
N Yesterday at 2:45 PM by melloncandy
I recently created an account; when I tried posting on the high school math forum, I received the error "New users are not allowed to post images in the community." Why is this? The post contains latex but has no images.
2 replies
principle9
Yesterday at 12:03 PM
melloncandy
Yesterday at 2:45 PM
k Classroom Problem
Joyful_Sapling   6
N Yesterday at 11:18 AM by Demetri
Hello guys!

I noticed something weird (it happened a lot), in the AoPS classroom, it wouldn't let me enter something twice. (Error image attached) I know this is something common, but if you want to say "no" for a question, and the next question your response is "no" as well, then you got to make it fancy so that it would let you send to the instructor. That is a problem. If a student doesn't want to waste a second making it fancy, then that is the problem. Maybe making it fancy would take so long that it might just make the person be less-possible to be pinned.

thank you
6 replies
Joyful_Sapling
Yesterday at 1:26 AM
Demetri
Yesterday at 11:18 AM
k Game Jam
shaayonsamanta   6
N Yesterday at 12:35 AM by shaayonsamanta
I can't join the Math Jam right now. Is anyone else experiencing this?

(It says I'm not enrolled in a class, which shouldn't matter for joining Math Jams.)
6 replies
shaayonsamanta
Yesterday at 12:28 AM
shaayonsamanta
Yesterday at 12:35 AM
k [RESOLVED] TeXeR Glitching
WalterMitchell   10
N Yesterday at 12:06 AM by WalterMitchell
When I was creating something in the AoPS TeXeR, and tried to output it as a bbcode, the "unrecognized server response" message popped up. Then, when I exported it as an image (a common trick to see the error), a very long script/error message came up. Lastly, when I updated a few things, this popped up (see below). Then, when I refreshed, my code was reset back to an earlier stage.

What happened, and is it fixable?

(i suspect this may be somewhat related to the extreme length of my asymptote script)

(EDIT: got another weird error message. attached)
10 replies
WalterMitchell
Tuesday at 8:35 PM
WalterMitchell
Yesterday at 12:06 AM
k FOr the win
hgauri   6
N Tuesday at 5:29 PM by A7456321
i cannot see the problem in FYI. What should I do?
6 replies
hgauri
Tuesday at 10:36 AM
A7456321
Tuesday at 5:29 PM
Error Connecting to Server
Zestra   9
N Jul 29, 2025 by ahxun2006
Hey guys just wanted to share an issue I've always had (looks like more site support threads are getting looked at so perfect timing)

Whenever I am in my AoPS ebooks, just on the web page I am guaranteed to get a notification that there is an "Error connecting to server. Check internet connection. We'll try again in 10s." In fact it just happened right now (total 3 times overall while writing) as I am writing this on the community page. It doesn't really affect me much when I am in the ebook since nothing much should be changing on there, but if I am looking at my feed while on the page, it can be problematic bc I dont get the messages once the error begins, and I have to refresh if I want to find out if I got any messages or new feed to look at. Pretty sure it happens like every 2 minutes for me.

It can't be just because of my internet connection but I move around a lot in the room and like to different places. And can still replicate.

Does anybody else have this issue?
9 replies
Zestra
Jul 28, 2025
ahxun2006
Jul 29, 2025
My TeXeR Is Not Saving, or Rendering.
AlienGirl05   5
N Jul 28, 2025 by IncredibleMongoose15
Greetings, Site Support,

I've been working on a TeXeR file. However, all of a sudden, it stopped saving my work, or rendering it. No matter how many times I pressed the save button, I wasn't being notified of my work being saved. Upon reloading the page, it wouldn't render my work (don't worry, I didn't actually lose any of my work; I have it saved in a backup Google Document). The "Render With BBCode" button isn't working either.

Thank you. :)
5 replies
AlienGirl05
Jul 28, 2025
IncredibleMongoose15
Jul 28, 2025
Graph Process Problem
Maximilian113   16
N Jul 23, 2025 by eduD_looC
Source: CMO 2025 P1
The $n$ players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.

The players will update their votes via a series of rounds. In one round, each player $a$ updates their vote, one at a time, according to the following procedure: At the time of the update, if $a$ is voting for $b,$ and $b$ is voting for $c,$ then $a$ updates their vote to $c.$ (Note that $a, b,$ and $c$ need not be distinct; if $b=c$ then $a$'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).

They repeat this updating procedure for $n$ rounds. Prove that at this time, all $n$ players will unanimously vote for the same person.
16 replies
Maximilian113
Mar 7, 2025
eduD_looC
Jul 23, 2025
Graph Process Problem
G H J
G H BBookmark kLocked kLocked NReply
Source: CMO 2025 P1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Maximilian113
592 posts
#1
Y by
The $n$ players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left.

The players will update their votes via a series of rounds. In one round, each player $a$ updates their vote, one at a time, according to the following procedure: At the time of the update, if $a$ is voting for $b,$ and $b$ is voting for $c,$ then $a$ updates their vote to $c.$ (Note that $a, b,$ and $c$ need not be distinct; if $b=c$ then $a$'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds).

They repeat this updating procedure for $n$ rounds. Prove that at this time, all $n$ players will unanimously vote for the same person.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Maximilian113
592 posts
#2
Y by
bruh
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
awesomeming327.
1788 posts
#3
Y by
Solution Sketch:

The idea is that there is this "center cycle" that will persist until it becomes a self-loop. Each time a member of the center cycle changes vote, the center cycle's size decreases by 1. One can find that this means that each time the procedure is run through, the center cycle decreases by at least half. This will take $\lfloor \log_2(n)\rfloor$

After that, we can consider distances to that self-loop. Each individual path decreases in a way very similar to that cycle we mentioned earlier, and so the worst case scenario is that the path size is $n-1$, whence it will take $\lfloor \log_2(n-1)\rfloor$

Summing these two together, one realizes that it is at most $n$.

My comments on the problem:

It is way, way, way too hard for a P1. This should've been put as P4 instead, maybe.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hectorleo123
368 posts
#4
Y by
I would say this is at least a P3 from the CMO.
$\textbf{Solution:}$

Let round \( 0 \) be the initial state (before any updates occur). Define \( G_i \) as the directed graph where the \( n \) players are the vertices, and the directed edges represent the votes after round \( i \), for all \( i \in \{0,1,\dots,n\} \).

$\textbf{Lemma:}$ \( G_i \) contains a unique cycle, and this cycle has length at most \( n - i \). Furthermore, any vertex outside the cycle is part of a directed path of length at most \( n - i \).

$\textbf{Proof:}$ We proceed by induction on \( i \).

$\textbf{Base case:}$ \( i = 0 \) (trivial).

$\textbf{Inductive Hypothesis:}$ Suppose the lemma holds for all \( i < k \leq n \).

$\textbf{Inductive Step:}$ Let \( v_1, v_2, \dots, v_t \) be the vertices of the cycle in \( G_{k-1} \) such that \( v_i \to v_{i+1} \) and \( v_{t+1} = v_1 \), where \( v_1 \) is the first vertex to update its vote in round \( k \).

Observe that we now have \( v_1 \to v_3 \), meaning that \( v_2 \) is no longer part of the cycle in \( G_k \). Notice that the only way a vertex \( v_j \) can leave the cycle and become part of a directed path (i.e., no longer in any cycle) is if there exists a vertex \( w \) in the cycle such that \( w \) votes for \( v_j \) and \( v_j \) previously belonged to the cycle...(\(\alpha\)).

Now, we verify that any directed path outside the cycle has length at most \( n - k \). By the inductive hypothesis, all such paths in \( G_{k-1} \) have length at most \( n - k + 1 \). In each round, the length of every path decreases by at least \( 2 \) from its outermost vertex towards the cycle, while it increases by at most \( 1 \) at the end touching the cycle (by (\(\alpha\))). Therefore, the length of any path is at most \( n - k \).

This completes the induction. Applying the lemma to \( G_n \), we conclude that all \( n \) players will eventually vote for the same person. $_\blacksquare$
This post has been edited 1 time. Last edited by hectorleo123, Mar 8, 2025, 12:38 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
genius_007
278 posts
#5
Y by
I don't think the solution above works fully, from your assumption that the first person to go lies in the cycle. We show how we can make a path grow by more than $2$ every round. Say we have a cycle $C$ and path $P$ coming off of $C$, where $|C| >> |P|$. Let the vertices in $C$ be $v_1,v_2,\dots,v_k$, where $v_i$ points to $v_{i+1}$. WLOG, let the first vertex $w$ of $P$ go to $v_1$. Update $w$ first, so it now points to $v_2$. Update $v_1$, so $v_1$ points to $v_3$. Update $v_2$, so $v_2$ points to $v_4$. Update $v_3$, so $v_3$ points to $v_5$. We may continue this, where $v_i$ points to $v_{i+2}$. In particular, we get the new path $w,v_2,v_4,\dots,v_{2r}$, where $r = \lfloor\dfrac{k}{2}\rfloor$. We've already updated all of those $r+1$ vertices in $P$, so $P$ has new length at least $r$. For large $C$, we are able to increase the length of $P$, which shows your solution is wrong.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hectorleo123
368 posts
#6
Y by
Sorry, my mistake, I meant that after the movement of v_1, v_2 left the cycle, then v_1 could disappear, fixed
This post has been edited 1 time. Last edited by hectorleo123, Mar 12, 2025, 9:55 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
cursed_tangent1434
743 posts
#7
Y by
Am I fakesolving or is the problem much easier than the above solutions indicate?

We prove the claim via induction. In the case where $n=2$ the result is trivial. Now, assume the result holds for some positive integer $k \ge 2$. When there are $k+1$ players, consider the very first update. Say this was an update for player $i$. Player $i$ votes for $i+1$ who votes for $i+2 \pmod{k+1}$ so $i$ will now vote for $i+2$. Note that then, there is no player voting for player $i+1$. Since each move replaces the player each player is voting for by some other player who is being voted for by some player, player $i+1$ will never receive a vote from any player ever again.

Now, player $i+1$ can be 'ignored'. He does not have any effect whatsoever on the players everyone else is voting for. However, the player he votes for changes over time (depending on the choices of the other players). Thus, each round can be considered as the conjunction of a round among $k$ players and an extra move for player $i+1$ (which may happen at any time during the round, in the first round it occurred as the first move). After $k$ rounds, players $1$ to $i$ and $i+2$ to $k+1$ will unanimously vote for the same person (who is within this set). Now, if player $i+1$ also votes for this player we are already done. Else, in the last round player $i+1$ adjusts his vote to match someone within the other set of $k$ players who all vote for the same person, which completes the induction.

Thus, repeating the updating procedure for $n$ rounds, always the $n$ players will end up voting for the same person as desired.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
genius_007
278 posts
#8
Y by
This is a fakesolve. After $i$ changes their vote to $i+1$, we cannot directly use the I.H. on $1,2,\dots,i,i+2,\dots,k+1$, as $i$ cannot have their vote updated (they already updated their vote).
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lw202277
4 posts
#9 • 2 Y
Y by swynca, MS_asdfgzxcvb
Treat the players and their votes as a directed graph $G$.

Claim: There is exactly 1 cycle $C$ in $G$.

This is initially true. Suppose $a \rightarrow b \rightarrow c$ becomes $a \rightarrow c$ and $b \rightarrow c$. The number of new cycles created is the number of directed paths from $c$ to $a$, and the number of cycles removed is the number of directed paths from $b$ to $a$. Since $b \rightarrow c$, this is equal to the number of directed paths from $c$ to $a$. Therefore the number of cycles stays constant.

Claim: The cycle's size decreases by at least 1 each round. (In hindsight, this was a very weak bound and can be replaced with $O(\log_2 n)$)

Just note that whenever a player $a \in C$ changes their vote from $b$ to $c$, $b$ is removed from the cycle unless $b=c$. If $b=c$ then we must have $a=b=c$ or else there would be no directed path from $b=c$ to $a$. This occurs iff the cycle has size 1.

Claim: Any branch (a directed path pointing into the cycle) has length at most 2, at the end of each round.

Note that any branch with length $\geq 2$ decreases its length when a player not voting for an element of $C$ switches their vote. It suffices to show that no branch can increase its length by more than 1 in any round.

Suppose $a \rightarrow b \rightarrow c$ where $a,b,c \in C$, and let $v_1 \rightarrow \dots \rightarrow v_k \rightarrow b$ be a branch. The only way for the branch to increase its length is for $b$ to be ejected from the cycle, which only occurs when $a$ switches their vote from $b$ to $c$. After this, $c$ is now the endpoint of the branch. Note that $c$ cannot be ejected from the cycle since this would require $a$ to switch their vote from $c$ to another player, which is impossible since $a$ already switched their vote. Thus the branch's length increases by at most 1 each round, which is nullified by it decreasing its length by at least 1 each round.

After round $n-1$ the cycle will have size 1, which is a player voting for themselves. All branches will have length at most 2, and each of these branches will reduce their length to 1 on round $n$. This implies that everyone will be voting for the same player. QED
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ray66
114 posts
#10
Y by
Treat the system as a group where elements consist of players and their respective votes. The group operation turns $a \rightarrow b \rightarrow c$ into $a \rightarrow c$ and $b \rightarrow c$.

$\textbf{Claim: }$ There exists only one cycle in each element

$\textbf{Claim: }$ Each round of operations decreases the length of the cycle by at least 1

$\textbf{Claim: }$ If the length of the cycle is $0$ we finish

$\textbf{Proof: }$ From the starting element (a circular graph) there is only 1 cycle. Each node in a 1-cycle graph can be characterized into nodes that comprise the cycle and nodes that feed into the cycle. The starting element only has nodes that comprise the cycle, and if $a$ and $b$ are both cycle nodes ($a$ and $b$ exist because the cycle length is at least 1), the operation $a \rightarrow b \rightarrow c$ into $a \rightarrow c$ and $b \rightarrow c$ turns $b$ into a feeding node while $a$ and $c$ remain cycle nodes.

$\textbf{Claim: }$ Feeding nodes always point into the cycle

$\textbf{Claim: }$ Each round of operations decrease the length of a feeding chain by at least 1

Because all processes must terminate and there exists exactly 1 cycle in each graph, everyone will eventually vote for the same player.
This post has been edited 1 time. Last edited by ray66, Mar 13, 2025, 3:13 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ru83n05
172 posts
#11
Y by
Here is how to achieve an $O(n)$ bound (which solutions above improve to $\lfloor\log_2 n\rfloor + 1$).

Call a player a leaf if no other player is voting for them. Once a leaf, they will always be a leaf. We claim that after $k\leq n-1$ rounds, there will be at least $k$ leaves.

We prove this by induction. In the first round, the first time a player updates their vote, the person they used to vote for becomes a leaf. Hence this is true when $k=1$.

Now, suppose we know the statement is true for $k$. If there exists a player that is only voted by leafs, that player will become a leaf at the end of the round. Otherwise, all non-leaf players are being voted by at least one non-leaf, and therefore exactly one non-leaf. In that case, the first time a non-leaf (in a nontrivial cycle) updates their vote, we get a vertex that is only voted by leafs. Hence by the end of the round we always get an extra leaf.

In particular, after $n-1$ rounds, all players will be voting for the same person.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
heheman
1099 posts
#13
Y by
how are u guys so genius bro teach me how i cant solve the first problem im gonna rage
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
genius_007
278 posts
#14 • 1 Y
Y by heheman
tbf this was arguably the hardest question on CMO this year, a lot of people were not able to solve
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
heheman
1099 posts
#15
Y by
guys, i think i got it!!!!

First draw graph with vertices representing players, and if $a$ votes for $b$ then there's arrow directed from $a$ to $b$. call the graph "connected" if the graph is connected when viewing the directed edges as undirected.

Observation 1: graph is always connected
Proof: note that every operation preserves it

Thus there is only ever 1 cycle in the whole graph, since if there is $2$ the only way it can exist if they are disjoint. not true by our observation.

Now note that the cycle decreases by at least $1$ for every turn. Note that during any turn it will always preserve the cycle structure. and at the end there are 2 arrows pointing to the same vertex by pigeonhole so one of them must have no arrow pointing to so it loses at least 1. note

anyways, let $k$ be the largest distance from a point traveling from outside the cycle needs to enter the cycle. Note that $k$ goes to $k-1+[$# of pts removed from the cycle}$]$ after every turn (or simply $k+[$# of pts removed from the cycle$]$ if $k$ was $0$ which is only at first turn, or $1$), at the worst case.

But every time some x = [# pts is removed from the cycle] is removed, it buys us $x-1$ turns of extra time in the future so we can subtract all of it from $k$

class ended so i am need to be brief
This post has been edited 1 time. Last edited by heheman, Jun 10, 2025, 10:47 PM
Reason: indexing error(was $0$ which is only at first turn -> was $0$ which is only at first turn, or $1$.)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
awesomeming327.
1788 posts
#16 • 1 Y
Y by heheman
I think most people who tried to do arguments of the form "cycle decreases by at least 1 each turn" got a 2/7

lol
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
heheman
1099 posts
#17
Y by
awesomeming327. wrote:
I think most people who tried to do arguments of the form "cycle decreases by at least 1 each turn" got a 2/7

lol

o sadge

do u know wut errors they might've made, idk if i fakesolve
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
eduD_looC
6611 posts
#18
Y by
The attached graph shows how the number of vertices with indegree 0 can stay constant after one round.
Attachments:
Z K Y
N Quick Reply
G
H
=
a