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

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

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

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

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0 replies
jwelsh
Jul 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
A nice identity
Synchrone   0
an hour ago
Source: Math&Maroc Competition 2025 Day1 Problem 3
For pairwise-distinct real numbers $a_1, \ldots, a_n$ prove that :
$$ \sum_{i =1}^n a_j^2 \prod_{k \neq j} \frac{a_j + a_k}{a_j - a_k} = (a_1 + \ldots + a_n)^2 $$
0 replies
Synchrone
an hour ago
0 replies
Moroccan Sets
Synchrone   0
an hour ago
Source: Math&Maroc Competition 2025 Day1 Problem 2
For any non-empty subsets $X$ and $Y$ of $\mathbb{N}_{\geq 1}$, we define $X \cdot Y := \{xy, x \in X, y \in Y\}$.
For instance, if : $$X = \{1,2,4\}, Y = \{3,4,6\}$$Then : $$ X \cdot Y = \{3,4,6,8,12,16,24\} $$We say that a subset $S$ of $\mathbb{N}_{\geq 1}$ is Moroccan if there exists two subsets $A$ and $B$ of $\mathbb{N}_{\geq 1}$ each containing at least two elements, such that $A \cdot B$ and $S$ are equal.
Prove that the set of perfect powers which are greater or equal to $2025$ is Moroccan
(A perfect power is an integer $n^k$, where $n > 1$ and $k > 1$ are integers)
0 replies
Synchrone
an hour ago
0 replies
Floor of a square root sequence
Synchrone   0
an hour ago
Source: Math&Maroc Competition 2025 Day 1 Problem 1
The sequence $(a_n)$ is defined by :
$$ a_1 =1, a_2 = 1, a_n = \lfloor \sqrt{2a_{n-1} + a_{n-2} + \ldots + a_1} \rfloor \text{ if } n > 2 $$Find $a_{2025}$
0 replies
Synchrone
an hour ago
0 replies
Question about Fubini integration in Euclidean spaces
euclides05   0
4 hours ago
Fubini's theorem in Lebesgue integration uses the product measure. My question is, how can this be applied in Euclidean spaces, where if $d= d_1+d_2$, the lebesgue measure of $R^d$ is the completion of the product measures
of $R^{d_1}$, $R^{d_2}$
and not the actual product measure?
I would be very pleased if someone provided a thorough answer cause i'm studying convolution stuff where it's used heavily and i haven't been able to find a stict explanation for it.
0 replies
euclides05
4 hours ago
0 replies
Bound on subsets
tenniskidperson3   33
N Jul 7, 2025 by quantam13
Source: 2012 USAMO problem #6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)

Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?
33 replies
tenniskidperson3
Apr 25, 2012
quantam13
Jul 7, 2025
1987 AMC 12 #26 - Probability
dft   5
N Jun 28, 2025 by superhuman233
The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}.$ Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$?

$ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{2}{5} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{3}{4} $
5 replies
dft
Jan 1, 2012
superhuman233
Jun 28, 2025
Square root and floor value and probability combo
Silverfalcon   5
N Jun 25, 2025 by mudkip42
Source: AHSME 1989 #20
Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor {\sqrt{x}} \rfloor = 12$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 120$. ($\lfloor {v} \rfloor$ means the greatest integer less than or equal to $v$.)

$\text{(A)} \ \frac{2}{25} \qquad \text{(B)} \ \frac{241}{2500} \qquad \text{(C)} \ \frac{1}{10} \qquad \text{(D)} \ \frac{96}{625} \qquad \text{(E)} \ 1$
5 replies
Silverfalcon
Dec 19, 2005
mudkip42
Jun 25, 2025
Probability of being divisible by 3
limac   20
N Jun 16, 2025 by mudkip42
Source: AMC 12B 2010, Problem 16
Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc + ab + a$ is divisible by $ 3$?

$ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$
20 replies
limac
Feb 25, 2010
mudkip42
Jun 16, 2025
Debra's Coin
ItsAmeYushi   26
N Jun 16, 2025 by superhuman233
Source: 2019 AMC 10 B #21
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

$\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$
26 replies
ItsAmeYushi
Feb 14, 2019
superhuman233
Jun 16, 2025
how to prepare AMC12 counting and number theory
lidabao8   0
Jun 11, 2025
Hi,

I am a 10th grader and i have already read the AOPS book of introduction to number theory + introduction to counting and probability, i have also enrolled in last year's AMC12 class as well this summer.

my biggest struggle is how to solve problems in number theory and counting in AMC12? the AOPS AMC12 class only has 1 session for each of these two topics, however these two topics are very wide and too many times when i am facing a problem i don't know where to start.

my goal is to qualify AIME via AMC12 so maybe aiming for 14-15 problems is sufficient. I feel the intermediate counting and probability is an overkill. could you please advise what else can I do to improve my skills in these 2 topics, such that next time when i read a problem i can have some clue where to start?

Thank you very much!
0 replies
lidabao8
Jun 11, 2025
0 replies
AMC 12 How to Study
krishiam   1
N Jun 9, 2025 by Alex-131
Hi, I am in my junior year and want to study for the upcoming AMC 12, but I'm not sure. I have done AOPS Geometry and Intermediate Algebra, but I forgot everything, and I have just a strong foundation in high school math and a basic understanding of probability, and know just the basic stuff needed. For reference, I scored like somewhere in the 70s AMC 10 last year. How should I prepare? Is grinding practice problems enough to qualify for AIME?
1 reply
krishiam
Jun 9, 2025
Alex-131
Jun 9, 2025
Die Problem
brainiac1   47
N Jun 8, 2025 by Kempu33334
Source: 2018 AIME II #13
Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
47 replies
brainiac1
Mar 23, 2018
Kempu33334
Jun 8, 2025
How To Improve
Challengees24   17
N Jun 8, 2025 by dragonlouis
I already know there is a ton of posts like this but something more tailormade for me would be nice.

Context: As a freshman, I got a 100.5 on the 10A and a 90 on the 10B smh. Not sugarcoating or anything it was a bad score.

So I want to improve in AMC and AIME and (hopefully) make JMO next year.

Firstly how much do you think I could improve, mostly I will study in the summer and breaks but schooldays are harder.

I have completed:
Intro to Pre-Alg
Intro To Alg.

I am in the middle of:
Volume 1
Intro To Counting and Probablity

Currently Own:
Volume 2
Intro To NT
Intro To Geo

Are there any specific classes, programs, summer things I should do, books i should start, end, buy, open to all feedback :thumbup:
17 replies
Challengees24
Nov 29, 2024
dragonlouis
Jun 8, 2025
Blackboard erasures: 2010 USAMO #6
tenniskidperson3   53
N Jun 4, 2025 by Mr.Sharkman
A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.
53 replies
tenniskidperson3
Apr 29, 2010
Mr.Sharkman
Jun 4, 2025
real analysis
ay19bme   1
N Apr 6, 2025 by alexheinis
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1 reply
ay19bme
Apr 6, 2025
alexheinis
Apr 6, 2025
real analysis
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ay19bme
287 posts
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alexheinis
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The limit equals $\int_0^1 \{1/x\}dx $ which we will calculate in another way.
We have $\int_{1/(n+1)}^1 \{1/x\}dx=\sum_1^n \int_{1/(k+1)}^{1/k} (1/x-[1/x])dx=\int_{1/(n+1)}^1 dx/x- \sum_1^n {1\over {k+1}}=$
$\ln(n+1)-\sum_2^{n+1} 1/k=1-(\sum_1^{n+1} 1/k-\ln(n+1))\rightarrow 1-\gamma$.
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