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k a My Retirement & New Leadership at AoPS
rrusczyk   1571
N Mar 26, 2025 by SmartGroot
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1571 replies
rrusczyk
Mar 24, 2025
SmartGroot
Mar 26, 2025
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

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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

To facilitate students in these efforts, we have created a new Peer-to-Peer Programs forum. With the creation of this forum, we are starting a new process for those of you who want to advertise your efforts. These advertisements and ensuing discussions have been cluttering up some of the forums that were meant for other purposes, so we’re gathering these topics in one place. This also allows students to find new peer-to-peer learning opportunities without having to poke around all the other forums.

To announce your program, or to invite others to work with you on it, here’s what to do:

1) Post a new topic in the Peer-to-Peer Programs forum. This will be the discussion thread for your program.

2) Post a single brief post in this thread that links the discussion thread of your program in the Peer-to-Peer Programs forum.

Please note that we’ll move or delete any future advertisement posts that are outside the Peer-to-Peer Programs forum, as well as any posts in this topic that are not brief announcements of new opportunities. In particular, this topic should not be used to discuss specific programs; those discussions should occur in topics in the Peer-to-Peer Programs forum.

Your post in this thread should have what you're sharing (class, session, tutoring, handout, math or coding game/other program) and a link to the thread in the Peer-to-Peer Programs forum, which should have more information (like where to find what you're sharing).
157 replies
jwelsh
Mar 15, 2021
cw357
Dec 11, 2023
k i C&P posting recs by mods
v_Enhance   0
Jun 12, 2020
The purpose of this post is to lay out a few suggestions about what kind of posts work well for the C&P forum. Except in a few cases these are mostly meant to be "suggestions based on historical trends" rather than firm hard rules; we may eventually replace this with an actual list of firm rules but that requires admin approval :) That said, if you post something in the "discouraged" category, you should not be totally surprised if it gets locked; they are discouraged exactly because past experience shows they tend to go badly.
-----------------------------
1. Program discussion: Allowed
If you have questions about specific camps or programs (e.g. which classes are good at X camp?), these questions fit well here. Many camps/programs have specific sub-forums too but we understand a lot of them are not active.
-----------------------------
2. Results discussion: Allowed
You can make threads about e.g. how you did on contests (including AMC), though on AMC day when there is a lot of discussion. Moderators and administrators may do a lot of thread-merging / forum-wrangling to keep things in one place.
-----------------------------
3. Reposting solutions or questions to past AMC/AIME/USAMO problems: Allowed
This forum contains a post for nearly every problem from AMC8, AMC10, AMC12, AIME, USAJMO, USAMO (and these links give you an index of all these posts). It is always permitted to post a full solution to any problem in its own thread (linked above), regardless of how old the problem is, and even if this solution is similar to one that has already been posted. We encourage this type of posting because it is helpful for the user to explain their solution in full to an audience, and for future users who want to see multiple approaches to a problem or even just the frequency distribution of common approaches. We do ask for some explanation; if you just post "the answer is (B); ez" then you are not adding anything useful.

You are also encouraged to post questions about a specific problem in the specific thread for that problem, or about previous user's solutions. It's almost always better to use the existing thread than to start a new one, to keep all the discussion in one place easily searchable for future visitors.
-----------------------------
4. Advice posts: Allowed, but read below first
You can use this forum to ask for advice about how to prepare for math competitions in general. But you should be aware that this question has been asked many many times. Before making a post, you are encouraged to look at the following:
[list]
[*] Stop looking for the right training: A generic post about advice that keeps getting stickied :)
[*] There is an enormous list of links on the Wiki of books / problems / etc for all levels.
[/list]
When you do post, we really encourage you to be as specific as possible in your question. Tell us about your background, what you've tried already, etc.

Actually, the absolute best way to get a helpful response is to take a few examples of problems that you tried to solve but couldn't, and explain what you tried on them / why you couldn't solve them. Here is a great example of a specific question.
-----------------------------
5. Publicity: use P2P forum instead
See https://artofproblemsolving.com/community/c5h2489297_peertopeer_programs_forum.
Some exceptions have been allowed in the past, but these require approval from administrators. (I am not totally sure what the criteria is. I am not an administrator.)
-----------------------------
6. Mock contests: use Mock Contests forum instead
Mock contests should be posted in the dedicated forum instead:
https://artofproblemsolving.com/community/c594864_aops_mock_contests
-----------------------------
7. AMC procedural questions: suggest to contact the AMC HQ instead
If you have a question like "how do I submit a change of venue form for the AIME" or "why is my name not on the qualifiers list even though I have a 300 index", you would be better off calling or emailing the AMC program to ask, they are the ones who can help you :)
-----------------------------
8. Discussion of random math problems: suggest to use MSM/HSM/HSO instead
If you are discussing a specific math problem that isn't from the AMC/AIME/USAMO, it's better to post these in Middle School Math, High School Math, High School Olympiads instead.
-----------------------------
9. Politics: suggest to use Round Table instead
There are important conversations to be had about things like gender diversity in math contests, etc., for sure. However, from experience we think that C&P is historically not a good place to have these conversations, as they go off the rails very quickly. We encourage you to use the Round Table instead, where it is much more clear that all posts need to be serious.
-----------------------------
10. MAA complaints: discouraged
We don't want to pretend that the MAA is perfect or that we agree with everything they do. However, we chose to discourage this sort of behavior because in practice most of the comments we see are not useful and some are frankly offensive.
[list] [*] If you just want to blow off steam, do it on your blog instead.
[*] When you have criticism, it should be reasoned, well-thought and constructive. What we mean by this is, for example, when the AOIME was announced, there was great outrage about potential cheating. Well, do you really think that this is something the organizers didn't think about too? Simply posting that "people will cheat and steal my USAMOO qualification, the MAA are idiots!" is not helpful as it is not bringing any new information to the table.
[*] Even if you do have reasoned, well-thought, constructive criticism, we think it is actually better to email it the MAA instead, rather than post it here. Experience shows that even polite, well-meaning suggestions posted in C&P are often derailed by less mature users who insist on complaining about everything.
[/list]
-----------------------------
11. Memes and joke posts: discouraged
It's fine to make jokes or lighthearted posts every so often. But it should be done with discretion. Ideally, jokes should be done within a longer post that has other content. For example, in my response to one user's question about olympiad combinatorics, I used a silly picture of Sogiita Gunha, but it was done within a context of a much longer post where it was meant to actually make a point.

On the other hand, there are many threads which consist largely of posts whose only content is an attached meme with the word "MAA" in it. When done in excess like this, the jokes reflect poorly on the community, so we explicitly discourage them.
-----------------------------
12. Questions that no one can answer: discouraged
Examples of this: "will MIT ask for AOIME scores?", "what will the AIME 2021 cutoffs be (asked in 2020)", etc. Basically, if you ask a question on this forum, it's better if the question is something that a user can plausibly answer :)
-----------------------------
13. Blind speculation: discouraged
Along these lines, if you do see a question that you don't have an answer to, we discourage "blindly guessing" as it leads to spreading of baseless rumors. For example, if you see some user posting "why are there fewer qualifiers than usual this year?", you should not reply "the MAA must have been worried about online cheating so they took fewer people!!". Was sich überhaupt sagen lässt, lässt sich klar sagen; und wovon man nicht reden kann, darüber muss man schweigen.
-----------------------------
14. Discussion of cheating: strongly discouraged
If you have evidence or reasonable suspicion of cheating, please report this to your Competition Manager or to the AMC HQ; these forums cannot help you.
Otherwise, please avoid public discussion of cheating. That is: no discussion of methods of cheating, no speculation about how cheating affects cutoffs, and so on --- it is not helpful to anyone, and it creates a sour atmosphere. A longer explanation is given in Seriously, please stop discussing how to cheat.
-----------------------------
15. Cutoff jokes: never allowed
Whenever the cutoffs for any major contest are released, it is very obvious when they are official. In the past, this has been achieved by the numbers being posted on the official AMC website (here) or through a post from the AMCDirector account.

You must never post fake cutoffs, even as a joke. You should also refrain from posting cutoffs that you've heard of via email, etc., because it is better to wait for the obvious official announcement. A longer explanation is given in A Treatise on Cutoff Trolling.
-----------------------------
16. Meanness: never allowed
Being mean is worse than being immature and unproductive. If another user does something which you think is inappropriate, use the Report button to bring the post to moderator attention, or if you really must reply, do so in a way that is tactful and constructive rather than inflammatory.
-----------------------------

Finally, we remind you all to sit back and enjoy the problems. :D

-----------------------------
(EDIT 2024-09-13: AoPS has asked to me to add the following item.)

Advertising paid program or service: never allowed

Per the AoPS Terms of Service (rule 5h), general advertisements are not allowed.

While we do allow advertisements of official contests (at the MAA and MATHCOUNTS level) and those run by college students with at least one successful year, any and all advertisements of a paid service or program is not allowed and will be deleted.
0 replies
v_Enhance
Jun 12, 2020
0 replies
k i Stop looking for the "right" training
v_Enhance   50
N Oct 16, 2017 by blawho12
Source: Contest advice
EDIT 2019-02-01: https://blog.evanchen.cc/2019/01/31/math-contest-platitudes-v3/ is the updated version of this.

EDIT 2021-06-09: see also https://web.evanchen.cc/faq-contest.html.

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
Is this FE solvable?
Mathdreams   0
26 minutes ago
Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
0 replies
Mathdreams
26 minutes ago
0 replies
OFM2021 Senior P1
medhimdi   0
32 minutes ago
Let $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ be two sequences of integers such that $a_{n+2}=a_{n+1}+a_n$ and $b_{n+2}=b_{n+1}+b_n$ for all $n\geq1$. Suppose that $a_n$ divides $b_n$ for an infinity of integers $n\geq1$. Prove that there exist an integer $c$ such that $b_n=ca_n$ for all $n\geq1$
0 replies
medhimdi
32 minutes ago
0 replies
Hard NT problem
tiendat004   2
N an hour ago by avinashp
Given two odd positive integers $a,b$ are coprime. Consider the sequence $(x_n)$ given by $x_0=2,x_1=a,x_{n+2}=ax_{n+1}+bx_n,$ $\forall n\geq 0$. Suppose that there exist positive integers $m,n,p$ such that $mnp$ is even and $\dfrac{x_m}{x_nx_p}$ is an integer. Prove that the numerator in its simplest form of $\dfrac{m}{np}$ is an odd integer greater than $1$.
2 replies
tiendat004
Aug 15, 2024
avinashp
an hour ago
disjoint subsets
nayel   2
N an hour ago by alexanderhamilton124
Source: Taiwan 2001
Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint.

what i have is this
2 replies
nayel
Apr 18, 2007
alexanderhamilton124
an hour ago
2012 AIME II Problem 12
xHypotenuse   4
N 4 hours ago by Soupboy0
Hello guys, I want to know what was wrong with my PIE approach.

First here's the problem:

For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.


My approach was finding the number of 7-unsafe numbers, 11-unsafe, and 13-unsafe, then finding 77-unsafe, 91-unsafe, 143-unsafe, and finally 1001-unsafe numbers and using PIE for a complementary counting approach. But somehow I got a number over than 10,000 for the total number of unsafe numbers. Is my approach valid and have I made arithmetic errors or does the PIE approach just not work?
4 replies
xHypotenuse
Yesterday at 11:40 PM
Soupboy0
4 hours ago
2026 AMC 8C
ChaitraliKA   2
N 6 hours ago by soumil2013
You’ve never seen an AMC 8 quite like this…

We, ChaitraliKA and Aaronjudgeisgoat, present
THE 2026 AMC 8C

Context

This is a totally legit mock test that you can use to predict what your real AMC 8 score will be!
We created many high quality problems for this test, and put our blood, sweat, and tears into it (and if u dont take it we’ll put your blood into it as well). We GUARANTEE that when you take a look at the problems, you will be stunned by the sheer quality oozing out of them. We know this for certain as we got this exact reaction from every single newborn baby that volunteered to take this test. This mock is perfect, not just for people looking to ace the AMC 8 next year, but even for high schoolers.

[quote=ChaitraliKA]
This was my first time making a mock, I wanted to make it as realistic as possible. As a person who has never taken an AMC 8 before, I thoroughly researched the types of problems and the format before making this. After some deep analysing, I have come to the conclusion that our mock highly resembles an AMC 8 with a -100% margin of error. I personally enjoyed making this, and put my heart and soul into making the problems and writing out the solutions. There was a lot of collaboration and every question has approval from both of us. As the most respected Cringe Nerdy Mathematician in history, I can guarantee you that it's not only about the math, it's about the experience of taking it. Please scroll down and take the test now. Thank you. [/quote]

[quote=Aaronjudgeisgoat]
After writing a MATHCOUNTS mock, I figured I would try and take it a step further - create an AMC 8 mock. As a fellow person who has never taken an AMC 8 before, I knew there was work needed to be done before I could accurately create one. I thoroughly reviewed the rules for this test, as well as the format. I even made sure to make every single answer choice viable, simulating the silliable nature of the actual AMC 8. We reviewed every single question, answer and solution, and after cross-comparing it with actual MAA exams, I found that it's not 100% accurate to the exam, but 200% accurate. That's right, it's accurate and even more accurate. Overall, this was a very fun experience, creating my first AMC mock so that people in need of math problems have something to work on. I thank @ChaitraliKA for giving me the opportunity to problem write, and I had a very fun time collaborating with the Cringiest and Nerdiest Mathematician of them all. In conclusion, I implore you to click the link below and proceed with the test, as not only will you learn, you will have fun.
Thank you for your blood.
[/quote]

You have two options on how to take it: Google Docs or MathDash
Google Docs:
Problems
Please read the guidelines carefully, and submit your answers for grading to ChaitraliKA and Aaronjudgeisgoat through AoPS PMs. We will send you the solutions as well, once we have graded you.
MathDash:
https://mathdash.fly.dev/contest/2026amc8c
If you do it on MathDash, we will still send you your final score through AoPS PMs, due to some issues.

We expect to start grading on April 15.

The leaderboard will be updated on the MSM post.
Good luck :)!


Please don't get this locked again. I will be very sad if that happens, because we put a lot of effort into this. If you're going to be a goody-two-shoes, just try to solve the legit problems and ignore the rest.

If you don't want to waste time, here are the legit problems that we would like you to try
2 replies
ChaitraliKA
Today at 1:14 PM
soumil2013
6 hours ago
[TEST RELEASED] Mock Geometry Test for College Competitions
Bluesoul   25
N Today at 8:46 AM by parmenides51
Hi AOPSers,

I have finished writing a mock geometry test for fun and practice for the real college competitions like HMMT/PUMaC/CMIMC... There would be 10 questions and you should finish the test in 60 minutes, the test would be close to the actual test (hopefully). You could sign up under this thread, PM me your answers!. The submission would close on March 31st at 11:59PM PST.

I would create a private discussion forum so everyone could discuss after finishing the test. This is the first mock I've written, please sign up and enjoy geometry!!

~Bluesoul

Discussion forum: Discussion forum

Leaderboard
25 replies
Bluesoul
Feb 24, 2025
parmenides51
Today at 8:46 AM
[March 2025] MEX MAGAZINE
Possible   1
N Today at 4:43 AM by Possible
Hello AoPS!

MathEXplained MAGAZINE aims to be a monthly newsletter that publishes articles in a newspaper-style format about various math related subjects in a way which can appeal to a broader audience. Topics may range from trending news stories within the math community to in-depth dissections of niche and interesting mathematical topics for more advanced readers.

Our Goal

While many of us on this platform might feel as if math is something super familiar to us, the truth is that this isn't the case outside of our tightly-knit community. Math doesn't get much coverage in mainstream media due to it being wrongfully seen by the general population either as a nerd's hobby or extremely foreign to the average person. Our mission as an organization is to promote mathematics through recognizing top mathletes, covering interesting math topics, and painting mathematics as something which can be fun and easy to get into.

March 2025 Edition

Celebrate the month of $\pi$ with MEX MAGAZINE's new publication! Check out this google drive file to solve some $\pi$ related math problems! Make sure to also read about our other spreads which cover Buffon's needle problem, the mathematics behind chess, and the history of $\pi$ day!

Website

If you are interested in reading our past issues, you can check out our website at this link. Over there, you can find links to our discord community where problem discussions can happen.

Staff

Our program is currently looking for new staff members! Please fill out this form to be considered.
1 reply
Possible
Today at 4:42 AM
Possible
Today at 4:43 AM
Nerfed 2023 JMO 1
brainfertilzer   29
N Today at 4:25 AM by Math-Lego
Source: 2024 AIME II P11
Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and
\[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.\]
29 replies
brainfertilzer
Feb 8, 2024
Math-Lego
Today at 4:25 AM
The 2nd geometry problem is #20
Frestho   59
N Today at 4:11 AM by megahertz13
Source: 2020 AMC 10A #20 / 2020 AMC 12A #18
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?

$\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$
59 replies
Frestho
Jan 31, 2020
megahertz13
Today at 4:11 AM
Tennessee Math Tournament (TMT) Online 2025
TennesseeMathTournament   62
N Today at 3:29 AM by TennesseeMathTournament
Hello everyone! We are excited to announce a new competition, the Tennessee Math Tournament, created by the Tennessee Math Coalition! Anyone can participate in the virtual competition for free.

The testing window is from March 22nd to April 12th, 2025. Virtual competitors may participate in the competition at any time during that window.

The virtual competition consists of three rounds: Individual, Bullet, and Team. The Individual Round is 60 minutes long and consists of 30 questions (AMC 10 level). The Bullet Round is 20 minutes long and consists of 80 questions (Mathcounts Chapter level). The Team Round is 30 minutes long and consists of 16 questions (AMC 12 level). Virtual competitors may compete in teams of four, or choose to not participate in the team round.

To register and see more information, click here!

If you have any questions, please email connect@tnmathcoalition.org or reply to this thread!

Thank you to our lead sponsor, Jane Street!

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62 replies
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TennesseeMathTournament
Mar 9, 2025
TennesseeMathTournament
Today at 3:29 AM
amc 10 prep
Aopsauser9999   3
N Today at 3:13 AM by Aopsauser9999
Source: hi
Hi! This year I got 69 and 72 (or something around those numbers) on the 2024 AMC 10A and 10B. I want to qualify for AIME this year. Is this a feasible goal? To prepare, should I do all of the exercises in Volume 1 and the intro books, then do mock tests and practice tests from mathdash and stuff?
3 replies
Aopsauser9999
Today at 2:34 AM
Aopsauser9999
Today at 3:13 AM
Mop Qual stuff
HopefullyMcNats2025   63
N Today at 2:41 AM by mathboy282
How good of an award/ achievement is making MOP, I adore comp math but am scared if I dedicate all my time to it I won’t get in a good college such as MIT or Harvard
63 replies
HopefullyMcNats2025
Sunday at 11:23 PM
mathboy282
Today at 2:41 AM
AMC 10/AIME Study Forum
PatTheKing806   106
N Today at 2:32 AM by BrocSoc
[center]

Me (PatTheKing806) and EaZ_Shadow have created a AMC 10/AIME Study Forum! Hopefully, this forum wont die quickly. To signup, do /join or \join.

Click here to join! (or do some pushups) :P

People should join this forum if they are wanting to do well on the AMC 10 next year, trying get into AIME, or loves math!
106 replies
PatTheKing806
Mar 27, 2025
BrocSoc
Today at 2:32 AM
the angle between any two lines is the same
mr.danh   3
N Jan 1, 2025 by kiyoras_2001
Source: Polish Mathematical Olympiad 2004 Final Round, Problem 5
Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.
3 replies
mr.danh
Feb 7, 2010
kiyoras_2001
Jan 1, 2025
the angle between any two lines is the same
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Source: Polish Mathematical Olympiad 2004 Final Round, Problem 5
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mr.danh
635 posts
#1 • 2 Y
Y by Adventure10, Mango247
Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.
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AshAuktober
953 posts
#2
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I believe the answer is 4 lines, for which we join the points in a regular tetrahedron to it's circumcentre.
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AshAuktober
953 posts
#3
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bump for proof that 5 doesn't work?
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kiyoras_2001
664 posts
#4
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The answer is actually $6$ and is reached for the major diagonals of icosahedron. See Miniature 9. Equiangular Lines in the book Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra by Jiřì Matoušek. There it is proved that the number of equiangular lines in $d$ dimensional space does not exceed $\binom{d+1}{2}$. However in dimensions higher than $3$ this bound is not tight.
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