Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
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
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
Number Theory Chain!
JetFire008   64
N 8 minutes ago by Primeniyazidayi
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
64 replies
JetFire008
Apr 7, 2025
Primeniyazidayi
8 minutes ago
equation 2025
mohamed-adam   0
24 minutes ago
Source: own
Find all positive integers $a,b$ such that $$a^8-2b^5=2025b$$
0 replies
mohamed-adam
24 minutes ago
0 replies
Serbian selection contest for the IMO 2025 - P1
OgnjenTesic   3
N 28 minutes ago by MR.1
Source: Serbian selection contest for the IMO 2025
Let \( p \geq 7 \) be a prime number and \( m \in \mathbb{N} \). Prove that
\[\left| p^m - (p - 2)! \right| > p^2.\]Proposed by Miloš Milićev
3 replies
OgnjenTesic
Yesterday at 4:01 PM
MR.1
28 minutes ago
Nice "if and only if" function problem
ICE_CNME_4   0
an hour ago
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
0 replies
ICE_CNME_4
an hour ago
0 replies
Proof Writing Help
gulab_jamun   1
N 3 hours ago by Gavin_Deng
Ok so like, i'm working on proofs, and im prolly gonna use this page for any questions. My question as of now is what can I cite? Like for example, if for a question I use Evan Chen's fact 5, in my proof do I have to prove fact 5 all over again or can i say "this result follows from Evan Chen's fact 5"?
1 reply
gulab_jamun
5 hours ago
Gavin_Deng
3 hours ago
for the contest high achievers, can you share your math path?
HCM2001   30
N 3 hours ago by mhgelgi
Hi all
Just wondering if any orz or high scorers on contests at young age (which are a lot of u guys lol) can share what your math path has been like?
- school math: you probably finish calculus in 5th grade or something lol then what do you do for the rest of the school? concurrent enrollment? college class? none (focus on math competitions)?
- what grade did you get honor roll or higher on AMC 8, AMC 10, AIME qual, USAJMO qual, etc?
- besides aops do you use another program to study? (like Mr Math, Alphastar, etc)?

You're all great inspirations and i appreciate the answers.. you all give me a lot of motivation for this math journey. Thanks
30 replies
HCM2001
May 21, 2025
mhgelgi
3 hours ago
Convolution of order f(n)
trumpeter   76
N 4 hours ago by ray66
Source: 2019 USAMO Problem 1
Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]for all positive integers $n$. Given this information, determine all possible values of $f(1000)$.

Proposed by Evan Chen
76 replies
trumpeter
Apr 17, 2019
ray66
4 hours ago
9 USAMO/JMO
BAM10   26
N Today at 1:36 PM by neeyakkid23
I mock ~90-100 on very recent AMC 10 mock right now. I plan to take AMC 10 final fives(9th), intermediate NT(9th), aime A+B courses in 10th and 11th and maybe mathWOOT 1 (12th). For more info I got 20 on this years AMC 8 with 3 sillies and 32 on MATHCOUNTS chapter. Also what is a realistic timeline to do this
26 replies
BAM10
May 19, 2025
neeyakkid23
Today at 1:36 PM
[TEST RELEASED] OMMC Year 5
DottedCaculator   110
N Today at 12:14 PM by PikaPika999
Test portal: https://ommc-test-portal-2025.vercel.app/

Hello to all creative problem solvers,

Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists? $\phantom{You lost the game.}$
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?

Check out the fifth annual iteration of the

Online Monmouth Math Competition!

Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.

Our website: https://www.ommcofficial.org/

This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.

How hard is it?

We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!

How are the problems?

You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc

How will the test be held?/How do I sign up?

Solo teams?

Test Policy

Timeline:
Main Round: May 17th - May 24th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.

Final Round: May 26th - May 28th
The top placing teams will qualify for this invitational round (5-10 questions). The final round consists of 5-10 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.

Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.

Scoring:

Prizes:

I have more questions. Whom do I ask?

We hope for your participation, and good luck!

OMMC staff

OMMC’S 2025 EVENTS ARE SPONSORED BY:

[list]
[*]Nontrivial Fellowship
[*]Citadel
[*]SPARC
[*]Jane Street
[*]And counting!
[/list]
110 replies
DottedCaculator
Apr 26, 2025
PikaPika999
Today at 12:14 PM
Proof-based math
imbadatmath1233   5
N Today at 3:19 AM by LearnMath_105
Okay, I need help in deciding on how i am going to prep. My JMO index was 121.5+11 = 231.5(10A) and I missed the cutoff by 1.5. Ive already grieved about this before but I need some help in deciding what I should do next year. I think I can make JMO but my goal is to get 21+ on JMO. However, OTIS applications are already done so does anyone have any other tips on how to prep for JMO. Any help would be very much appreciated. Also, how much time should i spend on computational if i want to prep for olympiad but I don't want to get rusty. Thanks for helping!
5 replies
imbadatmath1233
Yesterday at 11:06 PM
LearnMath_105
Today at 3:19 AM
Awesome Math Rec Letter
cowstalker   0
Today at 12:18 AM
Hello, I recently looked at the MIT Primes website and saw that they accept recommendation letters from the Awesome Math Summer Program. Has anyone ever gotten a recommendation letter from one of the teachers in Awesome Math? I'm also planning to take AMSP and would like to get a rec letter from my teacher, too, so I was wondering if this is even possible or not.
0 replies
cowstalker
Today at 12:18 AM
0 replies
Circles, Lines, Angles, Oh My!
atmchallenge   19
N Yesterday at 10:47 PM by kilobyte144
Source: 2016 AMC 8 #23
Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?

$\textbf{(A) }90\qquad\textbf{(B) }105\qquad\textbf{(C) }120\qquad\textbf{(D) }135\qquad \textbf{(E) }150$
19 replies
atmchallenge
Nov 23, 2016
kilobyte144
Yesterday at 10:47 PM
another diophantine about primes
AwesomeYRY   133
N Yesterday at 9:19 PM by EpicBird08
Source: USAMO 2022/4, JMO 2022/5
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.
133 replies
AwesomeYRY
Mar 24, 2022
EpicBird08
Yesterday at 9:19 PM
MAA finally wrote sum good number theory
IAmTheHazard   96
N Yesterday at 4:54 PM by megahertz13
Source: 2021 AIME I P14
For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$
96 replies
IAmTheHazard
Mar 11, 2021
megahertz13
Yesterday at 4:54 PM
Inequality with permutations
Vlados021   9
N Dec 14, 2024 by Seungjun_Lee
Source: 2019 Belarus Team Selection Test 7.3
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
(B. Serankou, M. Karpuk)
9 replies
Vlados021
Sep 2, 2019
Seungjun_Lee
Dec 14, 2024
Inequality with permutations
G H J
G H BBookmark kLocked kLocked NReply
Source: 2019 Belarus Team Selection Test 7.3
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vlados021
184 posts
#1 • 4 Y
Y by brokendiamond, Ab_Rin, Adventure10, farhad.fritl
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
(B. Serankou, M. Karpuk)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
brokendiamond
349 posts
#3 • 1 Y
Y by Adventure10
How to solve this problem ???
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
XbenX
590 posts
#5 • 1 Y
Y by Adventure10
We're trying to maximize $$C_n=\sum_{i=1}^{n}(a_i-b_i)^2=\sum_{i=1}^{2n}i^2-2\sum_{i=1}^{n}a_ib_i=\frac{2n(2n+1)(4n+1)}{6}-2\sum_{i=1}^{n}a_ib_i$$, so we need to minimize $S:=\sum_{i=1}^{n}a_ib_i$.

Lemma. The minimum of $S$ is achieved when none of the pairs $(a_i,b_i)$ are both greater than $n$.

Proof. Assume not, then there are four numbers $i,j,k,l \in \{1,2,3,\dots,n\}$ such that $(n+i)(n+j)+lk$ appears in $S$, but we can replace these numbers with $(n+i)l+(n+j)k$ and decrease $S$ because $(n+i)(n+j-l)\geq k(n+j-l)$. $\blacksquare$

So, we can let $A=\{1,2,\dots ,n\}$ and $B=\{n+1,n+2,\dots ,2n\}$ and by rearrangement inequality we get that $$S\geq \sum_{i=1}^{n} i(2n+1-i)=\frac{n(n+1)(2n+1)}{3}$$.
Hence, we have $C_n=\frac{2n(2n+1)(4n+1)}{6}-2S\leq \frac{n(2n+1)(2n-1)}{3}$.
This post has been edited 1 time. Last edited by XbenX, Jan 29, 2020, 1:10 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
somartino
8 posts
#6 • 2 Y
Y by Aegaertargaryen1102, YOUsername
I don't quite understand. We have to find the maximum value of the sum for each partition and then the smallest of those, don't we?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
indulged
69 posts
#8
Y by
XbenX wrote:
We're trying to maximize $$C_n=\sum_{i=1}^{n}(a_i-b_i)^2=\sum_{i=1}^{2n}i^2-2\sum_{i=1}^{n}a_ib_i=\frac{2n(2n+1)(4n+1)}{6}-2\sum_{i=1}^{n}a_ib_i$$, so we need to minimize $S:=\sum_{i=1}^{n}a_ib_i$.

Lemma. The minimum of $S$ is achieved when none of the pairs $(a_i,b_i)$ are both greater than $n$.

Proof. Assume not, then there are four numbers $i,j,k,l \in \{1,2,3,\dots,n\}$ such that $(n+i)(n+j)+lk$ appears in $S$, but we can replace these numbers with $(n+i)l+(n+j)k$ and decrease $S$ because $(n+i)(n+j-l)\geq k(n+j-l)$. $\blacksquare$

So, we can let $A=\{1,2,\dots ,n\}$ and $B=\{n+1,n+2,\dots ,2n\}$ and by rearrangement inequality we get that $$S\geq \sum_{i=1}^{n} i(2n+1-i)=\frac{n(n+1)(2n+1)}{3}$$.
Hence, we have $C_n=\frac{2n(2n+1)(4n+1)}{6}-2S\leq \frac{n(2n+1)(2n-1)}{3}$.
Oh,you did it in a wrong place ,you need to find where S max is,not the miniment。But your method is right ,and i think the ture result is n。
This post has been edited 1 time. Last edited by indulged, May 7, 2021, 2:22 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Takeya.O
769 posts
#9
Y by
$n$ is even: $\frac{13n^{3}-4n}{12}$
$n$ is odd: $\frac{13n^{3}-n}{12}$
This post has been edited 1 time. Last edited by Takeya.O, Jun 24, 2023, 7:12 AM
Reason: error
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Takeya.O
769 posts
#10
Y by
I don't know the time when the sum is minimized.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
dragon-YBW
40 posts
#11
Y by
use karamata ineq.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
dragon-YBW
40 posts
#12
Y by
Consruction:
if n=2k+1,take A={1,2,...,k,3k+2,...,4k+2} and B={k+1,...,3k+1}
if n=2k,take A={1,...,k,3k+1,...,4k} and B={k+1,...,3k}
Proof:
let c(k)=|a(k)-b(k)|,rearrange c(k) as d1≥d2≥...≥dn
if n=2k+1,we can prove that {dk}>{3k+1,3k,...,k+1}
if n=2k,we have {dk}>{3k-1,3k-1,3k-3,3k-3,...,k+1,k+1} (nontrivial but not hard)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Seungjun_Lee
526 posts
#13
Y by
I don't know why there isn't any full solution for this great problem! Nice and elegant integer inequality from 2019 RMMSL. XD

Let $s_i = a_i + b_i$. Since $\sum{a_i^2} + \sum{b_i^2}$ is constant, we only need to find the maximum value of $\sum_{i = 1}^{n} a_ib_i$ where $A$ and $B$ are partitions of $[2n]$, and $|A| = |B| = n$. This is simply maximizing $\sum_{i = 1}^{n} s_i^2$.

We claim that $|s_i - s_j| \le n$ for any $i < j$ both in the set $\{ 1, 2, \cdots, n\}$. This is because $|s_i - s_j| = |a_i - a_j + b_i - b_j|$. As $a_i \ge a_1 + i - 1$ and $a_j \le a_n - n + j$, we have $a_i - a_j \ge a_1 - a_n + (n-1) + (i - j)$. Also, $b_i - b_j \ge j - i$. Therefore, we have that $|s_i - s_j| \le n$. Now, we forget about our original definition of $s_i$, and just focus on the fact that $s_i$ are integers that satisfies $|s_i - s_j| \le n$ with $\sum_{i = 1}^{n} s_i = n(2n+1)$. As $|s_i - s_j| \le n$, we can set an integer $t$ so that $s_i \in [t, t+n]$ for any $i$. As the function $x \mapsto x^2$ is convex, we can force for some $s_i < s_j$, changing $(s_i, s_j)$ into $(s_i - 1, s_j + 1)$ makes the value $\sum_{i = 1}^{n} s_i^2$ larger. If there are two $i, j$ such that $a < s_i < s_j < a+n$, then we can force $s_i = a$ or $s_j = a+n$ so that the sum gets larger. Hence, we can see that only one value can lie in the interval $(t, t+n)$. The remaining job is just tedious calculation, which I will omit. :pilot:

Now, we proved that (I actually did the calculation) $\sum s_i^2$ is maximized when the sequence of $s_i$ consists of $\lfloor n/2 \rfloor$ number of $\left \lfloor \dfrac{3n}{2} \right \rfloor$ and $\lceil n/2 \rceil$ number of $\left \lceil \dfrac{5n}{2} \right \rceil$. Then, we can easily construct $A$ and $B$ so that $s_i$ becomes the prementioned sequence. Hence, we can calculate $C_n$, which turns out to be$$\sum_{i=1}^{n}{(a_i - b_i)^2} \ge \dfrac{2n(2n+1)(4n+1)}{3} - \left \lfloor \dfrac{n}{2} \right \rfloor \cdot \left \lfloor \dfrac{3n}{2} \right \rfloor^2 - \left \lceil \dfrac{n}{2} \right \rceil \cdot \left \lceil \dfrac{5n}{2} \right \rceil^2$$
This post has been edited 1 time. Last edited by Seungjun_Lee, Dec 14, 2024, 2:41 AM
Z K Y
N Quick Reply
G
H
=
a