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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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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
[TEST RELEASED] OMMC Year 5
DottedCaculator   158
N 10 minutes ago 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]
158 replies
+6 w
DottedCaculator
Apr 26, 2025
PikaPika999
10 minutes ago
EGMO (geo) Radical Center Question
gulab_jamun   0
32 minutes ago
For this theorem, Evan says that the power of point $P$ with respect to $\omega_1$ is greater than 0 if $P$ lies between $A$ and $B$. (I've underlined it). But, I'm a little confused as I thought the power was $OP^2 - r^2$ and since $P$ is inside the circle, wouldn't the power be negative since $OP < r$?
0 replies
gulab_jamun
32 minutes ago
0 replies
[CASH PRIZES] IndyINTEGIRLS Spring Math Competition
Indy_Integirls   67
N an hour ago by OGMATH
[center]IMAGE

Greetings, AoPS! IndyINTEGIRLS will be hosting a virtual math competition on May 25,
2024 from 12 PM to 3 PM EST.
Join other woman-identifying and/or non-binary "STEMinists" in solving problems, socializing, playing games, winning prizes, and more! If you are interested in competing, please register here![/center]

----------

[center]Important Information[/center]

Eligibility: This competition is open to all woman-identifying and non-binary students in middle and high school. Non-Indiana residents and international students are welcome as well!

Format: There will be a middle school and high school division. In each separate division, there will be an individual round and a team round, where students are grouped into teams of 3-4 and collaboratively solve a set of difficult problems. There will also be a buzzer/countdown/Kahoot-style round, where students from both divisions are grouped together to compete in a MATHCOUNTS-style countdown round! There will be prizes for the top competitors in each division.

Problem Difficulty: Our amazing team of problem writers is working hard to ensure that there will be problems for problem-solvers of all levels! The middle school problems will range from MATHCOUNTS school round to AMC 10 level, while the high school problems will be for more advanced problem-solvers. The team round problems will cover various difficulty levels and are meant to be more difficult, while the countdown/buzzer/Kahoot round questions will be similar to MATHCOUNTS state to MATHCOUNTS Nationals countdown round in difficulty.

Platform: This contest will be held virtually through Zoom. All competitors are required to have their cameras turned on at all times unless they have a reason for otherwise. Proctors and volunteers will be monitoring students at all times to prevent cheating and to create a fair environment for all students.

Prizes: At this moment, prizes are TBD, and more information will be provided and attached to this post as the competition date approaches. Rest assured, IndyINTEGIRLS has historically given out very generous cash prizes, and we intend on maintaining this generosity into our Spring Competition.

Contact & Connect With Us: Email us at indy@integirls.org.

---------
[center]Help Us Out

Please help us in sharing the news of this competition! Our amazing team of officers has worked very hard to provide this educational opportunity to as many students as possible, and we would appreciate it if you could help us spread the word!
67 replies
Indy_Integirls
May 11, 2025
OGMATH
an hour ago
Rational sequences
tenniskidperson3   58
N 6 hours ago by meduh6849
Source: 2009 USAMO problem 6
Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.
58 replies
tenniskidperson3
Apr 30, 2009
meduh6849
6 hours ago
4th grader qual JMO
HCM2001   38
N Today at 1:14 PM by blueprimes
i mean.. whattttt??? just found out about this.. is he on aops? (i'm sure he is) where are you orz lol..
https://www.mathschool.com/blog/results/celebrating-success-douglas-zhang-is-rsm-s-youngest-usajmo-qualifier
38 replies
HCM2001
May 22, 2025
blueprimes
Today at 1:14 PM
An FE. Who woulda thunk it?
nikenissan   120
N Today at 12:32 PM by NerdyNashville
Source: 2021 USAJMO Problem 1
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ \[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]
120 replies
nikenissan
Apr 15, 2021
NerdyNashville
Today at 12:32 PM
Zsigmondy's theorem
V0305   3
N Today at 9:00 AM by CatCatHead
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
3 replies
V0305
Yesterday at 6:22 PM
CatCatHead
Today at 9:00 AM
Base 2n of n^k
KevinYang2.71   50
N Today at 1:39 AM by ray66
Source: USAMO 2025/1, USAJMO 2025/2
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
50 replies
KevinYang2.71
Mar 20, 2025
ray66
Today at 1:39 AM
How Math WOOT Level 2 prepare you for olympiad contest
AMC10JA   0
Yesterday at 11:35 PM
I know how you do on Olympiad is based on your effort and your thinking skill, but I am just curious is WOOT level 2 is generally for practicing the beginner olympiad contest (like USAJMO or lower), or also good to learn for hard olympiad contest (like USAMO and IMO).
Please share your thought and experience. Thank you!
0 replies
AMC10JA
Yesterday at 11:35 PM
0 replies
Equilateral triangle $ABC$, $DEF$ has twice the area
v_Enhance   122
N Yesterday at 10:37 PM by lpieleanu
Source: JMO 2017 Problem 3, Titu, Luis, Cosmin
Let $ABC$ be an equilateral triangle, and point $P$ on its circumcircle. Let $PA$ and $BC$ intersect at $D$, $PB$ and $AC$ intersect at $E$, and $PC$ and $AB$ intersect at $F$. Prove that the area of $\triangle DEF$ is twice the area of $\triangle ABC$.

Proposed by Titu Andreescu, Luis Gonzales, Cosmin Pohoata
122 replies
v_Enhance
Apr 19, 2017
lpieleanu
Yesterday at 10:37 PM
Perfect Square Dice
asp211   67
N Yesterday at 9:27 PM by A7456321
Source: 2019 AIME II #4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
67 replies
asp211
Mar 22, 2019
A7456321
Yesterday at 9:27 PM
HCSSiM results
SurvivingInEnglish   75
N Yesterday at 7:25 PM by cowstalker
Anyone already got results for HCSSiM? Are there any point in sending additional work if I applied on March 19?
75 replies
SurvivingInEnglish
Apr 5, 2024
cowstalker
Yesterday at 7:25 PM
Perfect squares: 2011 USAJMO #1
v_Enhance   227
N Yesterday at 7:23 PM by ray66
Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.
227 replies
v_Enhance
Apr 28, 2011
ray66
Yesterday at 7:23 PM
Mustang Math Recruitment is Open!
MustangMathTournament   0
Yesterday at 7:02 PM
The Interest Form for joining Mustang Math is open!

Hello all!

We're Mustang Math, and we are currently recruiting for the 2025-2026 year! If you are a high school or college student and are passionate about promoting an interest in competition math to younger students, you should strongly consider filling out the following form: https://link.mustangmath.com/join. Every member in MM truly has the potential to make a huge impact, no matter your experience!

About Mustang Math

Mustang Math is a nonprofit organization of high school and college volunteers that is dedicated to providing middle schoolers access to challenging, interesting, fun, and collaborative math competitions and resources. Having reached over 4000 U.S. competitors and 1150 international competitors in our first six years, we are excited to expand our team to offer our events to even more mathematically inclined students.

PROJECTS
We have worked on various math-related projects. Our annual team math competition, Mustang Math Tournament (MMT) recently ran. We hosted 8 in-person competitions based in Washington, NorCal, SoCal, Illinois, Georgia, Massachusetts, Nevada and New Jersey, as well as an online competition run nationally. In total, we had almost 900 competitors, and the students had glowing reviews of the event. MMT International will once again be running later in August, and with it, we anticipate our contest to reach over a thousand students.

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MustangMathTournament
Yesterday at 7:02 PM
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Sums of pairs in a sequence
tenniskidperson3   56
N Apr 6, 2025 by Marcus_Zhang
Source: USAJMO 2010, Problem 2
Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties:
(a). $x_1 < x_2 < \cdots < x_{n-1}$ ;
(b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$;
(c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.
56 replies
tenniskidperson3
Apr 29, 2010
Marcus_Zhang
Apr 6, 2025
Sums of pairs in a sequence
G H J
Source: USAJMO 2010, Problem 2
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kamatadu
480 posts
#44 • 1 Y
Y by HoripodoKrishno
I claim that the only sequence is $x_i=2i$. It is easy to see that such a sequence suffices the condition. Now we prove that this is the only one.

Firstly, for some $n$, we prove that $x_i=i\cdot x_1$. We proceed using induction. The base case $x_1=1\cdot x_1=x_1$ is clearly true. Now assume the induction hypothesis for some $k-1<n-1$, and we prove it for $k$.

Consider the following sequence.\[x_1+x_{n-k},x_2+x_{n-k},\ldots,x_{k-1}+x_{n-k}.\]Note that this sequence has $k-1$ terms in it. Firstly, we have that $x_k+x_{n-k}=2n$ and the fact that $x_i<x_k$ for all $1\le i\le k-1$. So this gives us that $x_i+x_{n-k}<x_k+x_{n-k}=2n$ which means that we will get some $x_j$ for each $i$ such that $x_j=x_i+x_{n-k}$. Now clearly from the increasing condition of the sequence, we get that $j\ge n-k+1$. So each $x_j$ belongs to the following sequence.\[x_{n-(k-1)},x_{n-(k)},\ldots,x_{n-1}.\]So note that this sequence also has $k-1$ terms. So this forces that the equality occurs in each of the values of $x_j$ from both the sequences. So this gives that $x_1+x_{n-k}=x_{n-(k-1)}$, which further gives that $x_1+(2n - x_{n-(n-k)})=2n - x_{n-((n-k+1)}\implies x_1 - x_k = -x_{k-1}\implies x_1+x_{k-1}=x_k$. Now using our induction hypothesis, we get that $x_{k-1}=(k-1)x_1$ which finally gives $x_k=k\cdot x_1$. Now then, we finally get that $x_{n-1}=(n-1)x_1$. Putting this into $x_1+x_{n-1}=2n$, we get that $x_1=2$ and we are done.
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trk08
614 posts
#45
Y by
We claim the only possible sequence is if $x_i=2i$. This sequence clearly works, and we now prove it is the only one.//

First of all, consider $x_1+x_{n-2}$. As $x_1>0$, it has to be more than $x_{n-2}$, which implies that it must be $x_{n-1}$. As each of the following sums are in decreasing order, from top to bottom:
\[x_1+x_{n-2}\]\[x_1+x_{n-3}\]\[\dots\]\[x_1+x_1,\]and there are $n-2$ of them, we can say that:
\[x_1+x_{n-2}=x_{n-1}\]\[x_1+x_{n-3}=x_{n-2}\]\[\dots\]\[x_1+x_1=x_2.\]

Therefore, it is easy to see that $x_{i}=ix_1$. Therefore, $nx_1=2n$, or $x_1=2$, as desired.
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Shreyasharma
683 posts
#46
Y by
Note that we are given,
  • $x_1 < x_2 < \dots < x_{n-1}$
  • $x_1 + x_{n-1} = x_2 + x_{n-2} = x_3 + x_{n-3} = \dots = 2n$
From the final condition we find $x_1 + x_2$, $x_1 + x_3$, $\dots$, $x_1+x_{n-2}$ all belong to $(x_i)$. Clearly $x_1 + x_{n-2} = x_{n-1}$ or that $$x_{n-2} = x_{n-1} - x_1$$Then $x_1 + x_{n-3} = x_{n-2}$, which in turn gives, $$x_{n-3} = x_{n-1} - 2x_1$$and so on. Inducting downwards we find $$x_{n-1-k} = x_{n-1} - kx_1$$However then $x_1 = x_{n-1} - (n-2)x_1$ which gives $nx_1 = 2n$, or $x_1 = 2$. Now from our induction we find the sequence of $(x_i)$ are simply the even integers from $2$ to $2n - 2$.
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joshualiu315
2534 posts
#47
Y by
Consider the sequence $x_1+x_1, x_1+x_2, \dots, x_1+x_{n-1} = 2n$. Each of the $n-2$ terms before $2n$ are obviously less than $2n$, and they must be part of the sequence $x_1, x_2, \dots, x_{n-1}$. This means we must necessarily have

\begin{align*}
    x_1+x_1 &= x_2, \\
    x_1+x_2 &= x_3, \\
    \dots &\phantom{=} \\
    x_1+x_{n-2} &= x_{n-1}.
\end{align*}
Hence, $x_k = kx_1$, and plugging this in gives

\[x_1+(n-1)x_1=2n \implies x_1=2.\]
Thus, $x_n = \boxed{2n}$, which is easily checked to be true.
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dolphinday
1328 posts
#48
Y by
We find that
\[x_1 < x_1 + x_1 < x_1 + x_2 \dots x_1 + x_{n-2}\]are all in $x_i$, due to $(c)$, and since there are $n-1$ terms, this sequence must be equivalent to
\[x_1, x_2, \dots, x_{n-1}\]So, we can rewrite each term as $ix_1$.
$\newline$
\[x_1 + x_{n-1} = 2n\]can be rewritten as
\[x_1 + (n-1)x_1 = 2n \implies nx_1 = 2n \implies x_1 = 2\]Due to the fact that each term is equal to $ix_1$, this sequence is just $x_i = 2i$.
This post has been edited 1 time. Last edited by dolphinday, Jan 6, 2024, 11:58 AM
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shendrew7
799 posts
#49
Y by
Due to the third condition, we know the sequence $x_2, x_3, \ldots, x_{n-1}, 2n$ must correspond with the terms of
\[x_1+x_1 < x_1+x_2 < \ldots < x_1+x_{n-2} < x_1+x_{n-1} = 2n.\]
Thus we know $x_k = k \cdot x_1$ for $1 \leq k \leq n-1$, from which the second condition tells us our only solution is $\boxed{x_i = 2i}$. $\blacksquare$
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peppapig_
280 posts
#50
Y by
I goofed last time;

I claim that the only possible sequence is $2$, $4$, $6$, $\dots$, $2n-2$. Note that by problem conditions, we have that
\[x_1+x_{n-1}=2n,\]but we also have that
\[x_1+x_{n-2}=x_j,\]for some integer $j$. However, since $x_1<x_2<\dots<x_{n-1}$, we must have that $x_{n-2}=x_{n-1}-x_1$. Similarly, we get that
\[x_k=x_1+x_{n-3}<x_1+x_{n-2}=x_{n-1},\]meaning that $k=n-2$. Continuing this, we get that our sequence must be $x_1$, $2x_1$, $\dots$, $(n-1)x_1$, and since $x_1+x_{n-1}=2n$, we also get that $x_1$ must be equal to $2$. Therefore the only possible sequence is $2$, $4$, $6$, $\dots$, $2n-2$, which indeed works, finishing the problem.
This post has been edited 1 time. Last edited by peppapig_, Feb 24, 2024, 3:33 AM
Reason: Missed a $
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two_steps
102 posts
#51
Y by
is this correct? why is my solution different from everyone else lol

For any two terms $x_i < x_j$, notice that the following terms are also in the sequence
\[x_i, x_j \rightarrow 2n-x_j \rightarrow 2n-x_j+x_i \rightarrow x_j-x_i\]This is essentially a stronger version of the converse of condition 3. From here, note that we may prove that each element is a multiple of $x_1$ by induction. The only possible sequences are now $1,2,\dots, n-1$ and $2,4,\dots,2(n-1)$. The first sequence doesn't work, while the second one does.
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ItsBesi
147 posts
#52
Y by
My solution is different from others so did I fakesolve?

We separate the problem into $2$ cases which both are analogous.

Case 1. $n-\text{is even}$
Click to reveal hidden text

Case 2. $n-\text{is odd}$
Click to reveal hidden text

Since in both cases we have: $\{x_1,x_{k_1}, \dots , x_{k_{n-2}} \}=\{x_1,x_2, \dots, x_{n-1} \}$ we get:

$x_1=x_1 , x_2=x_{k_1} , x_3=x_{k-2} ,  \dots , x_{n-1}=x_{k_{n-2}}$
So
$x_1+x_1=x_{k_1}=x_2 \implies x_1+x_1=x_2 \implies x_2=2 \cdot x_1$
$x_1+x_2=x_{k_2}=x_3 \implies x_1+x_2=x_3 \implies x_3=x_1+x_2=x_1 + 2 \cdot x_1=3 \cdot x_1 \implies x_3=3 \cdot x_1$
$x_2+x_2=x_{k_3}=x_4 \implies x_2+x_2=x_4 \implies x_4=2 \cdot x_2=2 \cdot 2 \cdot x_1= 4 \cdot x_1 \implies x_4=4 \cdot x_1$
$x_2+x_3=x_{k_4}=x_5 \implies x_2+x_3=x_5 \implies x_5=x_2+x_3=2 \cdot x_1 + 3 \cdot x_1 =5 \cdot x_1 \implies x_5= 5 \cdot x_1$
$x_3+x_3=x_{k_5}=x_6 \implies x_3+x_3=x_6\implies x_6=x_3+x_3=3 \cdot x_1 + 3 \cdot x_1=6 \cdot x_1 \implies x_6=6 \cdot x_1$
$\dots$
$x_{\frac{n}{2}-1}+x_{\frac{n}{2}-1}=x_{k_{n-3}}=x_{n-2} \implies x_{\frac{n}{2}-1}+x_{\frac{n}{2}-1}=x_{n-2} \implies x_{n-2}= x_{\frac{n}{2}-1}+x_{\frac{n}{2}-1}=2 \cdot  x_{\frac{n}{2}-1} = 2 \cdot (\frac{n}{2}-1) \cdot x_1=(n-2) \cdot x_1 \implies x_{n-2}=(n-2) \cdot x_1$

Now again by $(b) \implies$

$x_2+x_{n-2}=2n \implies 2 \cdot x_1 + (n-2) \cdot x_1 =2n \implies n \cdot x_1=2n \implies x_1=2
So x_1=2, x_2=2 \cdot x_1=2 \cdot 2=4 , x_3=3 \cdot x_1= 3 \cdot 2=6 , \dots x_{n-1}=(n-1) \cdot x_1=(n-1) \cdot 2=2n-2$

Hence $(x_1,x_2, \dots , x_{n-1}=(2,4, \dots ,2n-2)$
This post has been edited 1 time. Last edited by ItsBesi, Sep 26, 2024, 1:12 PM
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Maximilian113
575 posts
#53
Y by
Observe that property (b) is equivalent to $$i+j < n \iff x_i+x_j < 2n.$$Therefore, $$x_1 < x_1+x_1 < x_1+x_2 < \cdots < x_1+x_{n-2} < 2n.$$However, each of these sums are some $x_k$ but since $$x_2<x_3<\cdots < x_{n-1} < 2n$$it follows that $x_1+x_k=x_{k+1}$ for each $k=1, 2, \dots, n-2.$ Thus $x_2=2x_1, x_3=3x_1,$ etc. This yields $2n=x_1+x_{n-1}=x_1+(n-1)x_1 \implies x_1=2,$ and it follows that the sequence is $$2, 4, 6, \cdots, 2n-2.$$It is easy to show that this works.
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eg4334
636 posts
#54
Y by
Goofy problem.

The conditions imply that the sums, in increasing order, $\{ x_1+x_1, x_1+x_2, \dots x_1+x_{n-1} \}$ must biject to $x_2, x_3, \dots x_{n-1}$. Therefore, $x_1+x_1=x_2$, $x_1+x_2=x_3$, and likewise. Then its not hard to see that $x_i = i x_1$ and from the second condition we obtain $x_1=2$. Therefore the only sequence is the increasing one of evens.
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Ihatecombin
66 posts
#55
Y by
Notice that \(a_1,a_1 + a_1, a_1 + a_2, \dots. a_1 +a_{n-2}\) are all different,
by an easy induction we obtain the only sequence that works is \(a_i = 2i\).
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bjump
1035 posts
#56 • 2 Y
Y by KenWuMath, imagien_bad
Solution from over a year ago ...

The answer is $x_{k}= 2n$
Note that
$$x_1 + x_{n-1}=2n$$$$x_{1}+ x_{n-2} = x_{n-1}$$$$\vdots$$$$x_{1}+x_{1}=x_{2}$$I claim that $x_{n} = nx_1$ we will prove this via induction with a trivial base case of $n=1$. Assume that this is true for $1 \le n \le k$ then $x_{k+1} = x_{k}+ x_1=kx_1+x_1= (k+1)x_1$. Now by condition (b) $x_i+x_{n-i} = 2n \iff (n-i+i)x_1=2n \iff x_1=2$. Therefore $x_k = k x_1 = 2k$.
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de-Kirschbaum
202 posts
#57
Y by
Note that $x_{n-2}<x_1+x_{n-2}<x_2+x_{n-2}=2n$ so by property 3 we must have $x_{n-2}+x_1=x_{n-1}$. By a similar reasoning this works for all $x_i$ so we have that $x_1, x_2,\ldots, x_{n-1}$ is an arithmetic sequence with common difference $x_1$, and by property 1 we must have $nx_1=2n \implies x_1=2$ so the only sequence that works is $x_i=2i$.
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Marcus_Zhang
980 posts
#58
Y by
Storage
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