School starts soon! Add problem solving to your schedule with our math, science, and/or contest classes!

G
Topic
First Poster
Last Poster
k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Aug 1, 2025
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

"As a parent, I'm deeply grateful to AoPS. Tiger has taken very few math courses outside of AoPS, except for a local Math Circle that doesn't focus on Olympiad math. AoPS has been one of the most important resources in his journey. Without AoPS, Tiger wouldn't be where he is today — especially considering he's grown up in a family with no STEM background at all."
— Doreen Dai, parent of IMO US Team Member Tiger Zhang

Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
CodeWOOT Code Jam - Monday, August 11th
ChemWOOT Chemistry Jam - Wednesday, August 13th
PhysicsWOOT Physics Jam - Thursday, August 14th
MathWOOT Math Jam - Friday, August 15th

There is still time to enroll in our last wave of summer camps that start in August at the Virtual Campus, our video-based platform, for math and language arts! From Math Beasts Camp 6 (Prealgebra Prep) to AMC 10/12 Prep, you can find an informative 2-week camp before school starts. Plus, our math camps don’t have homework and cover cool enrichment topics like graph theory. Our language arts courses will build the foundation for next year’s challenges, such as Language Arts Triathlon for levels 5-6 and Academic Essay Writing for high school students.

Lastly, Fall is right around the corner! 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. We’ve opened new Academy locations in San Mateo, CA, Pasadena, CA, Saratoga, CA, Johns Creek, GA, Northbrook, IL, and Upper West Side (NYC), New York.

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
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Aug 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
integral
teomihai   3
N 12 minutes ago by wipid98
Find $\int_{0}^{1}(fofofo...of)dx $
where $f(x)=x^3-\frac{3}{2}x^2+\frac{3}{4}$
3 replies
teomihai
Today at 12:41 PM
wipid98
12 minutes ago
limit function
aktyw19   4
N 14 minutes ago by P0tat0b0y
Find $\lim_{(x,y)\to (0,0)}=\frac{1-cos(x^2+y^2)}{(x^2+y^2)x^2y^2}$
4 replies
aktyw19
May 5, 2016
P0tat0b0y
14 minutes ago
Minimal number of edges
rompompole   0
an hour ago
Let $G$ be a bipartite graph on $2n + 1$ vertices with diameter $3$. If $\Delta(G) = n$, prove that $G$ has at least $3n -2$ edges.
0 replies
rompompole
an hour ago
0 replies
Prove direct sum in another way
SillinessSquared   0
2 hours ago
Source: Essential Linear Algebra, Titu Andresscu
(All equations written in Typst in a custom format then retyped in Latex; please excuse the odd syntax)

Let $V$ be a vector space over $F$ and $T: V --> V$ be a linear transformation such that $ \text{ker} T = \text{ker} T^2 $ and $Im T = Im T^2.$
Prove that $ V = \text{ker} T \oplus \text{Im} T. $

My friend solved this but he used the rank nullity theorem and didn't use the fact that Im T = Im T^2. I haven't learned that theorem, but I did learn about projections and their properties. Could someone help using the following definitions?

#example(number: [5.3])[#set enum(numbering : "a.")
7. We introduce a fundamental class of linear transformations: *projections onto subspaces.* Suppose $V$ is a vector space over a field $F$ and that $W_1,W_2$ are subspaces of $V$ such that $V = W_1 \oplus W_2$. The *projection onto $W_1$ along $W_2$* is the map $p: V -> W_1$ defined as follows: for each $v \in V, p(v) \in W_1: v - p(v) \in W_2$.

]

#theorem(number: [5.15])[#set enum(numbering : "a)")
Let $V$ be a vector space over a field $F$ and let $T: V --> V$ be a linear map on $V$. The following statements are equivalent:
+ $T$ is a projection
+ We have $T \circ T = T$. Moreover, if this is the case, then $\text{ker} T \oplus \text{Im} (T) = V$.
]
0 replies
SillinessSquared
2 hours ago
0 replies
MathGuass 2025 Summer Math Contest
MathandPhysics-Life   4
N 4 hours ago by MathandPhysics-Life
Hello AoPS community!

I'm reposting this announcement because my original post seems to have been removed, and I'm not sure why. My apologies if this causes any confusion(or if this is not fine, please let me know)!

I'm excited to announce the first-ever MathGauss Contest! This is a free online contest featuring 10 questions to be completed in a 45-minute time limit. The problems are designed for students preparing for competitions, with a level ranging from AMC 10 to AIME.

You can find all the contest details and problems here: https://mathgauss.com/contest
https://mathgauss.com/contest

Good luck to all the participants!
4 replies
MathandPhysics-Life
Aug 1, 2025
MathandPhysics-Life
4 hours ago
9 NSF results and general feedback
vmene   2
N 5 hours ago by GP102
Source: Own
NSF is a Indian-based foundation designed to raise money for kids and commence competitions to let students learn in a creative way. This is just my opinion. source And please post which event you got the place in. I got $3^{\text{rd}}$ place in MB2 and $10^{\text{th}}$ place in ISC. :) :coolspeak: :ddr:
2 replies
vmene
Today at 12:41 PM
GP102
5 hours ago
Non optimal Tic Tac Toe
fidgetboss_4000   23
N Today at 11:59 AM by Ash_the_Bash07
Source: AMC 12A #22
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the $3$-by-$3$ array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all $9$ boxes are filled or one of the players has $3$ of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?

$\textbf{(A)}\ 36 \qquad\textbf{(B)}\  112 \qquad\textbf{(C)}\  120 \qquad\textbf{(D)}\
 148 \qquad\textbf{(E)}\ 160$
23 replies
fidgetboss_4000
Nov 11, 2021
Ash_the_Bash07
Today at 11:59 AM
POV: You're a student
fidgetboss_4000   18
N Today at 11:59 AM by Ash_the_Bash07
Source: AMC 12A #7
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5, $ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$?

$\textbf{(A)}\ {-}18.5  \qquad\textbf{(B)}\  {-}13.5 \qquad\textbf{(C)}\  0 \qquad\textbf{(D)}\
 13.5 \qquad\textbf{(E)}\ 18.5$
18 replies
fidgetboss_4000
Nov 11, 2021
Ash_the_Bash07
Today at 11:59 AM
Divisor Function
aopsuser305   35
N Today at 11:40 AM by mpcnotnpc
Source: 2021 AMC10A #23, 2021 AMC12A #20
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$

$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
35 replies
aopsuser305
Nov 11, 2021
mpcnotnpc
Today at 11:40 AM
Favorite Problems from the 2024-2025 School Year
djmathman   46
N Today at 7:44 AM by xoonks_rbo_sus
[center]
IMAGE
[/center]

As many people (whether from the competitor side or the problem-writing side) get ready for the 2025-2026 school year, we often try to find nice and interesting problems to solve and hone our skills. Many of these problems are classics which have passed through years or generations. But this often leaves us to throw problems from recent years to the sides. While not all problems from any given year are great, there are some that do deserve praise and are undeservedly overlooked.

So let's fix this! The question: what were some of your favorite problems from competitions that occurred in the 2024-2025 school year?

Just to clarify: this is not necessarily asking for problems which you may have solved this year for practice. Instead, I'm talking about e.g. problems which came from contests hosted within the past 12 months - so for example AMC/AIME/USA(J)MO, ARML, PUMaC, HMMT, CMIMC, CHMMC, Duke, etc., as well as any national olympiads or TST's from this year. (2024 ISL problems also count since they were released to the public in 2025.) Problems from mocks released this year are eligible, too. Also, to make things easier to read, I highly recommend writing the original problem statements in your posts, as opposed to just stating the sources.

I also think this is a good way to officially end the 2024-2025 school year of contests. :)

I'm interested in seeing what some of your responses are!
46 replies
djmathman
Jul 23, 2025
xoonks_rbo_sus
Today at 7:44 AM
USAJMO, USAMO, AIME, ARML, HMMT, Mandelbrot, PUMaC,USAMTS Mocks collected
parmenides51   33
N Today at 5:19 AM by parmenides51
Source: Usamo , Usajmo, AIME Mocks
2 years have passed since I posted geometry problems from mock USAJMO and USAMO collected. Last days I started posting in HSO the Mock Contests of USAJMO level, not posted there or not existing already inside User Created Contests


So enjoy

$\bullet$ USAJMO Mocks posted in separate threads, collected in post collections geometry addicted may look here
$\bullet$ USAMO Mocks, posted in separate threads, collected in post collections geometry addicted may look here

$\bullet$ AIME Mocks geometry addicted may look here
$\bullet$ ARML Mocks geometry addicted may look here
$\bullet$ HMMT Mocks
$\bullet$ Mandelbrot Mocks
$\bullet$ PUMaC Mocks
$\bullet$ USAMTS Mocks

$\bullet$ Different Type Mocks

$\bullet$ USA Computational Geo Mocks (from AMC, AIME, HMMT, PUMaC Mocks)
$\bullet$ Geometry from USA Computational Mocks (AIME + ARML + Other Computational + Different Type)

$\bullet$ Geometry from AIME Mocks + 3D Geo from Mock AIMEs + Conics from Mock AIMEs
$\bullet$ Geometry from AIME (official)

Collected among others inside here, among USA Mocks

after I finish with USAJMO Mocks, I shall start doing the same with USAMO Mocks (a few are here ) (more soon)

if you happen to come across other USAJMO mocks, please let me know to create their post collections, to include them in the list above

enjoy / start solving

only 1 old USAJMO Mock remain to be posted

aops mock AIME wiki 2012-23 almost complete
33 replies
parmenides51
Nov 17, 2023
parmenides51
Today at 5:19 AM
Difficulty of needle questions in AOPS Volume 1
yobro1froggyjr   2
N Today at 3:14 AM by NamelyOrange
I'm very new to competition math and recently purchased AOPS Vol 1, where I have been working through the problems. In this book, some questions are marked with a needle, indicating that they are more difficult than the other questions.

What is the general difficulty of these problems? (AOPS Competition rating) My guess is around a 1.5.
2 replies
yobro1froggyjr
Yesterday at 6:43 PM
NamelyOrange
Today at 3:14 AM
9 Most recommended resource
HiCalculus   28
N Today at 1:14 AM by Happyllamaalways
For this poll, I'm going to let everyone choose 1 option to vote for so I can see which resource is the absolute best. Later, I'll make another poll with 2 options per user to find the best way to incorporate 2 resources to study for the test in around 3 months from now.
28 replies
HiCalculus
Jul 27, 2025
Happyllamaalways
Today at 1:14 AM
NSF Nats 2025
Vkmsd   33
N Today at 12:48 AM by Vkmsd
Is anyone going to North South Foundation's national finals this year?

(For those who don't understand NSF is like an Indian organization that runs contests to raise funds for scholarships in India.)
33 replies
Vkmsd
Jul 16, 2025
Vkmsd
Today at 12:48 AM
Continuous functions
joybangla   3
N Apr 7, 2025 by Rohit-2006
Source: Romanian District Olympiad 2014, Grade 11, P2
[list=a]
[*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
functions. Prove that $f$ is also continuous.
[*]Give an example of a discontinuous function $f\colon\mathbb{R}
\rightarrow\mathbb{R}$, with the following property: there exists an interval
$I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a}
\colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]
3 replies
joybangla
Jun 15, 2014
Rohit-2006
Apr 7, 2025
Continuous functions
G H J
G H BBookmark kLocked kLocked NReply
Source: Romanian District Olympiad 2014, Grade 11, P2
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
joybangla
836 posts
#1 • 3 Y
Y by Adventure10, Mango247, and 1 other user
  1. Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
    $g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
    $h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
    functions. Prove that $f$ is also continuous.
  2. Give an example of a discontinuous function $f\colon\mathbb{R}
\rightarrow\mathbb{R}$, with the following property: there exists an interval
    $I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a}
\colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aziiri
1640 posts
#2 • 2 Y
Y by Adventure10, Mango247
$g(2x)=f(2x)+f(4x)=g(x)-f(x)+h(x)-f(x)$ therefore : $f(x) =\frac{h(x)+g(x)-g(2x)}{4}$, since $g,h$ are continuous we get that $f$ is continuous too.
I have a question for the second, is $f$ discontinuous at every point of $\mathbb{R}$ ?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mavropnevma
15142 posts
#3 • 3 Y
Y by Adventure10, Mango247, and 1 other user
No, it is enough to exhibit an example where $f$ has just one point of discontinuity. Moreover, the interval $I$ must not be degenerated (to a point).

As an example, the signum function $f(0)=0$ and $f(x) = x/|x|$ for $x\neq 0$, with $I=(-\infty,0)$, for which $g_a$ is identically null for any $a\in I$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Rohit-2006
250 posts
#4
Y by
Brothers and Sisters of AOPS, I am going to give my solution .....
Part 1:
We express $f(x)$ in terms of $g(x)$ and $h(x)$ and since $g(x)$ is continuous so $g(2x)$ is also continuous. So
$$f(x)=\frac{g(x)+h(x)-g(2x)}{2}$$and hence $f(x)$ is continuous.
Part 2:
\[
f(x) = \begin{cases} 
1, x\geq 0\\
-1, x<0
\end{cases} 
\]
Z K Y
N Quick Reply
G
H
=
a