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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

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

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

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

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

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

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

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

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

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

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

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

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

Advertising paid program or service: never allowed

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

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

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

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
Inspired by my own results
sqing   2
N 7 minutes ago by lbh_qys
Source: Own
Let $ a,b,c\geq 2.$ Prove that
$$ (a+1)(b+2)(c +1)-3 abc\leq 12$$$$ (a+1)(b+2)(c +1)-\frac{7}{2}abc\leq  8$$$$ (a+1)(b+3)(c +1)-\frac{15}{4}abc\leq  15$$$$ (a+1)(b+3)(c +1)-4abc\leq  13$$
2 replies
sqing
2 hours ago
lbh_qys
7 minutes ago
Function Equation
Dynic   0
9 minutes ago
Find all $f:\mathbb{R} \to \mathbb{R}$ such that
$$f(x-f(y))=f(x+f(y)+y^5)+f(2f(y)+y^5)+2025,\forall x,y\in \mathbb{R}$$
0 replies
Dynic
9 minutes ago
0 replies
Cubic function from Olymon
Adywastaken   0
an hour ago
Source: Olymon Volume 11 2010 663
Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$x^2y^2(f(x+y)-f(x)-f(y))=3(x+y)f(x)f(y)$ $\forall$ $x,y \in \mathbb{R}$
0 replies
+1 w
Adywastaken
an hour ago
0 replies
Sequence bounding itself
juckter   2
N an hour ago by Math-Problem-Solving
Source: Own
We say a sequence of integers $a_1, a_2, \dots, a_n$ is self-bounded if for each $i$, $1 \le i \le n$ there exist at least $a_i$ terms of the sequence that are less than or equal to $i$. Find the maximum possible value of $a_1 + a_2 + \dots + a_n$ for a self-bounded sequence $a_1, a_2, \dots, a_n$.
2 replies
juckter
Jan 13, 2021
Math-Problem-Solving
an hour ago
F=ma USAPhO qualification
RabtejKalra   5
N 2 hours ago by torch
Which would be more benificial to USAPhO qualification, doing the AoPS physics courses (Intro to Physics all the way to F=ma Problem Series) or doing the first half of HRK?
5 replies
RabtejKalra
Yesterday at 10:56 PM
torch
2 hours ago
USA Canada math camp
Bread10   20
N 2 hours ago by torch
How difficult is it to get into USA Canada math camp? What should be expected from an accepted applicant in terms of the qualifying quiz, essays and other awards or math context?
20 replies
Bread10
Mar 2, 2025
torch
2 hours ago
high tech FE as J1?!
imagien_bad   52
N 2 hours ago by Maximilian113
Source: USAJMO 2025/1
Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective.
Note: A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.
52 replies
imagien_bad
Yesterday at 12:00 PM
Maximilian113
2 hours ago
Weird DeMoivre Stuff
EGMO   19
N 3 hours ago by Magnetoninja
Source: 2023 AMC 12A P25
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that
$$
\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}
$$whenever $\tan 2023x$ is defined. What is $a_{2023}?$

$\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023$
19 replies
EGMO
Nov 9, 2023
Magnetoninja
3 hours ago
questions from a first-time applicant to math camps
akliu   23
N 3 hours ago by John_Mgr
hey!! im a first time applicant for a lot of math camps (namely: usa-canada mathcamp, PROMYS, Ross, MathILY, HCSSiM), and I was just wondering:

1. how much of an effect would being a first-time applicant have on making these math camps individually?
2. I spent a huge amount of effort (like 50 or something hours) on the USA-Canada Mathcamp application quiz in particular, but I'm pretty worried because supposedly almost no first-time applicants get into the camp. Are there any first-time applicants that you know of, and what did their applications (as in, qualifying quiz solutions) look like?
3. Additionally, a lot of people give off the impression that not doing the full problem set will screw your application over, except in rare cases. How much do you think a fakesolve would impact my PROMYS application chances?

thanks in advance!
23 replies
akliu
Mar 12, 2025
John_Mgr
3 hours ago
Anyone LFT for SMT?
Mathdreams   0
4 hours ago
Hi everyone,

Is there anyone willing to join an SMT (Stanford Math Tournament) team?

I have a team looking for one more person.

Edit: If you are interested, please PM me, and I'll answer any questions there :)
0 replies
Mathdreams
4 hours ago
0 replies
MOHS for Day 1
MajesticCheese   16
N 4 hours ago by KevinChen_Yay
What is your opinion for MOHS for day 1?

JMO 1:
JMO 2/AMO 1:
JMO 3:
AMO 2:
AMO 3:
16 replies
MajesticCheese
Yesterday at 3:15 PM
KevinChen_Yay
4 hours ago
Average score for USA/Canada Mathcamp Applicant
duwoah   7
N 4 hours ago by Aarush12
Hi I'm applying to USA/Canada mathcamp and just want to know how I'm doing compared to others. How many problems on average does a typical accepted applicant solve?
7 replies
duwoah
Feb 21, 2025
Aarush12
4 hours ago
USAJMO 2025
hashbrown2009   8
N 4 hours ago by KevinChen_Yay
Predict what you got for first 3 in 2025 USAJMO.

I predict I got:

(6)(6)(5) so 17.
8 replies
hashbrown2009
5 hours ago
KevinChen_Yay
4 hours ago
Base 2n of n^k
KevinYang2.71   37
N 4 hours ago by MathLuis
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 $d$.
37 replies
KevinYang2.71
Yesterday at 12:01 PM
MathLuis
4 hours ago
Minimum number of values in the union of sets
bnumbertheory   4
N Oct 16, 2023 by math90
Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3
For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
4 replies
bnumbertheory
Oct 14, 2023
math90
Oct 16, 2023
Minimum number of values in the union of sets
G H J
G H BBookmark kLocked kLocked NReply
Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3
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bnumbertheory
14 posts
#1
Y by
For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
Z K Y
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This post has been deleted. Click here to see post.
alexheinis
10478 posts
#2
Y by
First we derive some small values and then we use two lemmata to show that $f(n)=n+2$ for $n\ge 3$.
We call a collection of sets good if the condition of the problem is satisfied, hence no inclusions and distinct cardinalities.
- we have $f(1)=0$ if you consider this case.
- suppose $A_1,A_2\subset [1,2]$ is good . Then the $|A_k|\in \{1,2\}$ are distinct, hence one of them equals 2. Contradiction. With $\{1\},\{2,3\}$ we have $f(2)=3$.
- suppose $A_1,A_2,A_3\subset [1,4]$ is good. Then the $|A_k|\in \{1,2,3\}$ are distinct, hence wlog $A_3=\{4\}$ and $A_1,A_2\subset \{1,2,3\}$ with cardinalities 2,3. Hence one of them is $\{1,2,3\}$, contradiction. With $\{1,4\},\{2,3,4\},\{5\}$ we see that $f(3)=5$.

Lemma 1. Suppose $A_1,\cdots, A_n\subset [1,n+2]$ are good with cardinalities $1,\cdots,n$. Then we can construct the same for $n+2$. Proof: let $B_k:=A_k\cup \{n+3\}, B_{n+1}:=[1,n+2],B_{n+2}=\{n+4\}$.

With induction base $n=2,3$ we find that $f(n)\le n+2$ for all $n\ge 2$.

Lemma 2. Suppose that $n\ge 3$ and $f(n)\le n+1$. Then also $f(n-1)\le n$.
Proof: suppose that $A_1,\cdots,A_n\subset [1,n+1]$ is good. The $|A_k|\in [1,n]$ are distinct hence wlog $A_n=\{n+1\}$. Then $A_1,\cdots, A_{n-1}\subset [1,n]$ are good.

Now suppose that $f(n)\le n+1$ for some $n\ge 3$. Then $n>3$ and we can use Lemma 2 to obtain $f(n-1)\le n$ etc until $f(3)\le 4$. Contradiction, hence $f(n)=n+2$ for $n\ge 3$.
This post has been edited 2 times. Last edited by alexheinis, Oct 15, 2023, 8:37 PM
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math90
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Above: nice solution. BTW, you can prove that $f(n)\ge n+2$ for $n\ge 3$ directly: suppose that $A_1,\cdots,A_n\subset [1,n+1]$ is good. Then since $A_i\not\subset A_j$ and $|A_i|\ne |A_j|$ for $i\ne j$ and $n\ge 3$, the sizes of $A_i$ are $1,\ldots,n$. Hence we can assume WLOG $A_n=\{n+1\}$. Then $A_1,\cdots, A_{n-1}\subset [1,n]$ are good of sizes $2,\ldots,n$. WLOG $A_{n-1}=\{1,\ldots,n\}$. But then $A_{n-2}\subset A_{n-1}$ (here we use the assumption $n\ge 3$), contradiction.
This post has been edited 4 times. Last edited by math90, Oct 16, 2023, 10:17 PM
Reason: Thanks @below
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alexheinis
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@math90: Thank you. What you write is nearly correct, we don't have inclusions in a good collection hence $A_i\subset A_j$ is not true when $i<j$. The cardinalities are distinct, however, and that's why we have $1,\cdots,n$ for the cardinalities. The rest of your argument is fine.
This post has been edited 1 time. Last edited by alexheinis, Oct 16, 2023, 9:33 PM
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math90
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Thanks, corrected now.
This post has been edited 1 time. Last edited by math90, Oct 16, 2023, 10:16 PM
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