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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/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.

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0 replies
jlacosta
Jun 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
Polynomial characteristif of matrix
M4tchash3l   1
N an hour ago by alexheinis
Suppose $A \in M_n(\mathbb{C})$ where $M_n(\mathbb{C})$ is a set of matrix size $n \times n$ with complex entries. Let $p(x)$ be polynomial characteristic of $A$ and $q(x) \in \mathbb{C}[x]$ is monic polynomial such that $\deg q = \deg p$ and $q(A)=0$. Is $q(x) = p(x)$?
1 reply
M4tchash3l
2 hours ago
alexheinis
an hour ago
Banach space
ILOVEMYFAMILY   1
N an hour ago by alexheinis
Let $$
c_{00} := \left\{ x = (x_1, x_2, \ldots, x_n, \ldots) : x_n \in \mathbb{R},\ x_n = 0\ \text{for all sufficiently large } n \right\}.
$$Equip \( c_{00} \) with the usual addition and scalar multiplication to make it into a vector space over \( \mathbb{R} \).
Does there exist a norm on \( c_{00} \) that makes it a Banach space?
1 reply
ILOVEMYFAMILY
Today at 12:25 AM
alexheinis
an hour ago
AMC 8 past comps
VivaanKam   3
N 6 hours ago by daniil
Hi, I am practicing for the 2025-2026 AMC 8 comp and want to try some of the problems from past comps can I find them on the AoPS website? If not, where?
3 replies
VivaanKam
Yesterday at 4:52 PM
daniil
6 hours ago
geo equals ForeBoding For Dennis
dchenmathcounts   102
N 6 hours ago by cxsmi
Source: USAJMO 2020/4
Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$.

Milan Haiman
102 replies
dchenmathcounts
Jun 21, 2020
cxsmi
6 hours ago
Titu Factoring Troll
GoodMorning   78
N Today at 3:33 AM by cxsmi
Source: 2023 USAJMO Problem 1
Find all triples of positive integers $(x,y,z)$ that satisfy the equation
$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$
78 replies
GoodMorning
Mar 23, 2023
cxsmi
Today at 3:33 AM
Putnam 1960 B4
sqrtX   4
N Today at 2:45 AM by megarnie
Source: Putnam 1960
Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.
4 replies
sqrtX
Jun 11, 2022
megarnie
Today at 2:45 AM
JOIN THIS COMPETITION
fatunicorn_22   3
N Today at 1:27 AM by fatunicorn_22
Join the CountHerInPDX Fall Math Competition, a free and fun online event just for girls in grades 3–8! Whether you're new to math contests or already love problem-solving, this competition is all about building confidence, making friends, and showing what girls can do in STEM. Sign up now to challenge yourself, win prizes, and be part of a community where girls lead in math!

https://sites.google.com/bsd48.org/countherinpdx/competition








3 replies
fatunicorn_22
Jun 19, 2025
fatunicorn_22
Today at 1:27 AM
f <= 2x+1 and xf(x+1) <= f(f+2)
truongphatt2668   2
N Today at 1:27 AM by truongphatt2668
Is there any function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfied:
$$f(x) \le 2x + 1, \forall x > 0$$$$xf(x+1) \le f(x)(f(x)+2), \forall x > 0$$
2 replies
truongphatt2668
Saturday at 12:47 PM
truongphatt2668
Today at 1:27 AM
27th ELMO 2025
DottedCaculator   115
N Today at 12:44 AM by RaymondZhu
27th ELMO on AoPS: Error Littered Math Olympiad / Elmo Likes Swapping Math Olympiads
Saturday, June 14th and Saturday, June 21st, 2025

The ELMO is an olympiad test similar to both the USAMO and IMO in format. It is written, administered, and graded by returning MOPers for those attending MOP for the first time. However, because there are only finitely many people who can attend MOP each year, for many years the competition has also been posted and run on AoPS, similarly to many of the other mocks that run on this site. Here are links to the previous AoPS ELMO threads.

You can also find the problems and shortlist in the USA contests section of the Contest Collections. The acronym is different every year and was chosen during the first week of MOP.

[list][*]You should sign up in this thread if you intend to do the contest. Signups are neither mandatory nor binding, but we want to have an estimate of how many people take the test.
[*]You may participate in either or both of the two days. Each day will consist of three problems of a similar difficulty to the USAMO. The top scores from each day and overall will be recognized, and all scores will be posted on the ELMO website, where you can also find past results.
[*]The Day 1 and Day 2 problems will be released in the afternoon on Saturday, June 14th and Saturday, June 21st, 2025, respectively.
[*]Submissions will be due on Thursday, June 26th at 11:59 PM EDT. You may submit for both days at once or separately.
[*]We want to encourage early submissions. Early submissions will receive priority in grading.
[*]Each day should be taken in a contiguous 4.5-hour period, similarly to how you would take the USAMO, although you do not need a proctor and you may take the test anytime before the submission deadline.
[*]Submit your day 1 solutions to here and your day 2 solutions to here. Please submit a separate PDF for each problem and include your username (or real name, if you want that to appear instead) and problem number in the filename. You may either scan written solutions using a scanner or phone app (e.g. Dropbox, CamScanner, Genius Scan, iOS Notes app, Adobe Scan), or write your solutions in LaTeX (compiled to PDF). You may write your solutions on paper and then transcribe them to LaTeX immediately afterwards, provided that you do so entirely verbatim. Please only do so if you have hideous handwriting that only you can decipher or do not have anything resembling a scanner available.
[*]After the tests are over, the problems will be posted in the High School Olympiads forum. Please do not discuss the problems with anyone until this happens.
[*]More information will be provided later!
[/list]
We look forward to your participation!
115 replies
DottedCaculator
Jun 14, 2025
RaymondZhu
Today at 12:44 AM
4th grader qual JMO
HCM2001   60
N Today at 12:43 AM by RaymondZhu
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
60 replies
HCM2001
May 22, 2025
RaymondZhu
Today at 12:43 AM
9 When do you start your AMC 10/12 prep?
ethan2011   18
N Today at 12:37 AM by giratina3
I am wondering when it is a good time to start prepping for AMC's, given that I am studying much more for olympiads this year rather than focusing on computational that much, and when I should stop doing a ton of proof questions/OTIS and start locking in on AMC's.
18 replies
ethan2011
Jun 11, 2025
giratina3
Today at 12:37 AM
2n equations
P_Groudon   85
N Yesterday at 5:37 PM by eg4334
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:

\begin{align*}
a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\
a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\
a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\
&\vdots & &\vdots \\
a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1}
\end{align*}
85 replies
P_Groudon
Apr 15, 2021
eg4334
Yesterday at 5:37 PM
FREE Online Math Camp This Summer!
lprado   1
N Yesterday at 5:21 PM by meduh6849
Lubbock Online Math Camp (LOMC)
AMC 8 & MATHCOUNTS Training for Middle Schoolers

Are you a middle school student who loves math? Do you want to improve your problem-solving skills and excel in future math competitions? Join this FREE online math camp this summer, designed to help prepare for MATHCOUNTS, AMC 8, and other middle school math contests.

When:
Every Sunday & Thursday, 6-7pm Central Time
Starting July 6th 2025, continuing through the rest of the year!

The Class:
Interactive Zoom classes to review problems and learn key math concepts.
Weekly homework, consisting of problem sets curated by me, as well as past contests.
Access to a Google Classroom for assignments and other resources.
Join a Discord community to talk with peers and make friends.
The syllabus will be given to students. The concepts covered will include prime factorization, triangles, Shoelace Theorem, Simon’s Favorite Factoring Trick, and more.
All sessions are recorded.

The instructors:
The class will be taught by Lishan Prado, Seonho Choi, and Eric Chen. We're students at Lubbock High School, and we love math! We've been participating in math contests for years, performing well in MATHCOUNTS, AMCs, and AIME. We're experienced competition math teachers, teaching at our local middle school's MATHCOUNTS program. If you have any questions, feel free to reach out.

Interest Form: https://forms.gle/BoKCysSgw7Rj7M5XA
Sign up before room runs out! We may limit the size of the class in order to preserve the quality of the camp.

Contact Me:
Email: lishan.prado@gmail.com
Discord: bluewater16
1 reply
lprado
Yesterday at 5:03 PM
meduh6849
Yesterday at 5:21 PM
Maximum visits in Planar National Park
ppanther   31
N Yesterday at 5:04 PM by eg4334
Source: USAMO 2021/2
The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)
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A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?
31 replies
ppanther
Apr 15, 2021
eg4334
Yesterday at 5:04 PM
Linear algebra problem
Feynmann123   1
N May 25, 2025 by Etkan
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
1 reply
Feynmann123
May 25, 2025
Etkan
May 25, 2025
Linear algebra problem
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Feynmann123
4 posts
#1
Y by
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
Z K Y
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Etkan
1581 posts
#2 • 1 Y
Y by Mathzeus1024
Feynmann123 wrote:
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.

a) If $\lambda \in \mathbb{C}$ is an eigenvalue of $A$ then there exists $v\in \mathbb{R}^n$ such that $v\neq 0$ and $Av=\lambda v$, so\begin{align*}A^2v & =A(Av) \\
& =A(\lambda v) \\
& =\lambda Av \\
& =\lambda (\lambda v) \\
& =\lambda ^2v.
\end{align*}Hence from $A^2=A$ we get $A^2v=Av$, and so $\lambda ^2v=\lambda v$. Since $v\neq 0$, this gives $\lambda ^2=\lambda$, and so $\lambda =0$ or $\lambda =1$.
You can also solve this using the Cayley-Hamilton Theorem, but let's keep things elementary.

b) Indeed, $A$ is a projection.

c) An example of such a $2\times 2$ matrix is $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$.
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