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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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k a 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:
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, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
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Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
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Prealgebra 2 Self-Paced

Prealgebra 2
Tuesday, Mar 25 - Jul 8
Sunday, Apr 13 - Aug 10
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21


Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
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Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
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Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Monday, Mar 17 - Jun 9
Thursday, Apr 17 - Jul 3
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Sunday, Mar 2 - Jun 22
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
Wednesday, Apr 23 - Oct 1
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Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
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Intermediate Counting & Probability
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Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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Wednesday, Apr 9 - Sep 3
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Sunday, Mar 23 - Jun 15
Wednesday, Apr 16 - Jul 2
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
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Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
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Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
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AMC 12 Problem Series
Tuesday, May 27 - Aug 12
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Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
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Intermediate Programming with Python
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USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
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Physics

Introduction to Physics
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Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
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Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
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
Stanford Math Tournament (SMT) Online 2025
stanford-math-tournament   5
N 39 minutes ago by stanford-math-tournament
[center]Register for Stanford Math Tournament (SMT) Online 2025[/center]


[center] :surf: Stanford Math Tournament (SMT) Online is happening on April 13, 2025! :surf:[/center]

[center]IMAGE[/center]

Register and learn more here:
https://www.stanfordmathtournament.com/competitions/smt-2025-online

When? The contest will take place April 13, 2025. The pre-contest puzzle hunt will take place on April 12, 2025 (optional, but highly encouraged!).

What? The competition features a Power, Team, Guts, General, and Subject (choose two of Algebra, Calculus, Discrete, Geometry) rounds.

Who? You!!!!! Students in high school or below, from anywhere in the world. Register in a team of 6-8 or as an individual.

Where? Online - compete from anywhere!

Check out our Instagram: https://www.instagram.com/stanfordmathtournament/

Register and learn more here:
https://www.stanfordmathtournament.com/competitions/smt-2025-online


[center]IMAGE[/center]


[center] :surf: :surf: :surf: :surf: :surf: [/center]
5 replies
stanford-math-tournament
Mar 9, 2025
stanford-math-tournament
39 minutes ago
AIME Math History
hashbrown2009   82
N 2 hours ago by stjwyl
Idk why but I wanted to see how good ppl are
Post all your AIME scores ever (if you qualified for USA(J)MO, you may put that score, too)

(Note: Please do not post fake scores. I legit want to see how good ppl are and see how good I am)
I'll start:

5th grade: AIME : 2 lol
6th grade: AIME : 5
7th grade: AIME : 8
8th grade : AIME : 13 USAJMO: 18
9th grade (rn): AIME: 11 (sold)
82 replies
1 viewing
hashbrown2009
Feb 20, 2025
stjwyl
2 hours ago
Power of 4: 2011 USAMO #4, 2011 USAJMO #6
tenniskidperson3   129
N 4 hours ago by chenghaohu
Consider the assertion that for each positive integer $n\geq2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of $4$. Either prove the assertion or find (with proof) a counterexample.
129 replies
tenniskidperson3
Apr 28, 2011
chenghaohu
4 hours ago
AIME score for college apps
Happyllamaalways   50
N 5 hours ago by Equinox8
What good colleges do I have a chance of getting into with an 11 on AIME? (Any chances for Princeton)

Also idk if this has weight but I had the highest AIME score in my school.
50 replies
Happyllamaalways
Thursday at 1:34 AM
Equinox8
5 hours ago
AMC 10 to AMC 12 Score Translation
MathRook7817   6
N 6 hours ago by MathRook7817
Hey guys, what does a 132 on the AMC 10 translate to on the AMC 12?
6 replies
MathRook7817
Yesterday at 6:44 PM
MathRook7817
6 hours ago
USPHO qual
jlcong   26
N Yesterday at 7:00 PM by jlcong
If I am consistently able to mock a 7, how can I improve in 3 easy steps.

Also will improving my amc10 score indirectly help my on my physics endeavors?
I know I can do it!!!
26 replies
jlcong
Feb 19, 2025
jlcong
Yesterday at 7:00 PM
[~$2000 In Prizes!] Register for Crack the Code: A Cybersecurity Hackathon
ayushiayanna   0
Yesterday at 6:59 PM
Hello!

We invite you to compete at Crack the Code, a defensive cybersecurity hackathon taking place from March 22nd-23rd at Concordia University, Irvine!.

We’re excited to offer you the opportunity to take part in our hackathon and compete in teams of 2-4 for ~$2000 in prizes, secure internships with professors and CEOs, and create a meaningful project. There is no cost to sign up and you will be provided with a lively working environment, fun side competitions, and most of all, FREE FOOD!

SIGN UP by filling out this form: bit.ly/crackthecode2025! For more information, see our website cui2025.crackthecode.dev or email us at cui@crackthecode.dev. Hope to see you there!

Happy Hacking,

The Crack the Code Team
0 replies
ayushiayanna
Yesterday at 6:59 PM
0 replies
2016 Sets
NormanWho   107
N Yesterday at 6:18 PM by ItsBesi
Source: 2016 USAJMO 4
Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set ${1, 2,...,N}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.
107 replies
NormanWho
Apr 20, 2016
ItsBesi
Yesterday at 6:18 PM
Vertices of a pentagon invariant: 2011 USAMO #2
tenniskidperson3   50
N Yesterday at 6:09 PM by blueprimes
An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.
50 replies
tenniskidperson3
Apr 28, 2011
blueprimes
Yesterday at 6:09 PM
9 Can I make MOP
Bigtree   14
N Yesterday at 3:58 PM by GlitchyBoy
My dream is to be on IMO team ik thats not going to happen b/c the kids that make it are like 6th mop quals :play_ball:. I somehow got a $208.5$ index this yr (118.5 on amc10+ 9 on AIME) i’m in 7th rn btw first year comp math also. I will grind so hard until like 30 hrs/week. I’m ok at proofs.
14 replies
Bigtree
Mar 9, 2025
GlitchyBoy
Yesterday at 3:58 PM
Geo equals ABsurdly proBEMatic
ihatemath123   72
N Yesterday at 7:15 AM by Ilikeminecraft
Source: 2024 USAMO Problem 5, JMO Problem 6
Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE = EC$. Let $M$ be the midpoint of $BC$.

Show that line $AB$ is tangent to the circumcircle of triangle $BEM$.

Proposed by Anton Trygub
72 replies
ihatemath123
Mar 21, 2024
Ilikeminecraft
Yesterday at 7:15 AM
average FE
KevinYang2.71   73
N Yesterday at 6:42 AM by Ilikeminecraft
Source: USAJMO 2024/5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]for all $x,y\in\mathbb{R}$.

Proposed by Carl Schildkraut
73 replies
KevinYang2.71
Mar 21, 2024
Ilikeminecraft
Yesterday at 6:42 AM
happy configs
KevinYang2.71   60
N Yesterday at 6:38 AM by Ilikeminecraft
Source: USAJMO 2024/2
Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$. A configuration of $mn$ rectangles is called happy if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.

Proposed by Serena An and Claire Zhang
60 replies
KevinYang2.71
Mar 20, 2024
Ilikeminecraft
Yesterday at 6:38 AM
[Registration open!] SCMC Spring 2025
Bluesoul   4
N Yesterday at 6:09 AM by Bluesoul
[center]IMAGE[/center]

Registration is open until March 14! Find the form here.

Calling all SoCal high-school math students!
From the University of Southern California, the student-run Southern California Mathematics Competition (SCMC) returns for another year!

Contest info
We plan to hold our Spring 2025 contest on Saturday, 29 March, on USC's University Park Campus. This time we are going to introduce the online version! Competitors not in Socal area could select the online version.

However, the online competition will be considered unofficial; i.e. only onsite, in-person competitors will be eligible for awards, but we might still recognize the top scorers. Online competitors will compete in the Individual Round only. We will send out the Team Round problems after the conclusion of the Individual Round, but we will not be collecting answers to these problems or scoring them.

[list]
[*] The intended target audience for this competition is high-school students, but middle-school students are also welcome to participate!
[*] As is typical, our competition will feature both individual and team-based components: you may choose to compete individually or as part of a team of 4 students, although we strongly encourage you to compete with a team.
[*] This year we are going to have one division only. All competitors will compete in a 20-problem, 120-minute individual round with problems on topics in algebra, number theory, combinatorics, and geometry.
[*] Those competitors on a team will then compete in a 10-problem, 30-minute team round. One or more problems on the team round will feature content covered in a faculty lecture given on the day of the competition.
[*] Before the awards ceremony, we will have a head-to-head MATHCOUNTS Countdown Round-style contest (just for fun). Top 8 competitors face off head-to-head in single-elimination tournament.
[/list]
Detailed information about scoring, timing, etc. can be found on our website.

Additionally, you may view samples of our past work here. If you're looking for resources to study, this page may also be of help.

Hope to see you there, and math on!
- Bluesoul, SCMC Competition Design Chair
4 replies
Bluesoul
Feb 19, 2025
Bluesoul
Yesterday at 6:09 AM
AIME Qualifier T-shirt?
xHypotenuse   39
N Yesterday at 3:37 AM by RainbowSquirrel53B
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online
39 replies
xHypotenuse
Mar 3, 2025
RainbowSquirrel53B
Yesterday at 3:37 AM
AIME Qualifier T-shirt?
G H J
G H BBookmark kLocked kLocked NReply
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xHypotenuse
732 posts
#1 • 1 Y
Y by HWenslawski
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online
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AD314159
17 posts
#2 • 1 Y
Y by HWenslawski
Where do u get the AIME qual tshirt?
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Pengu14
395 posts
#3 • 1 Y
Y by HWenslawski
AD314159 wrote:
Where do u get the AIME qual tshirt?

Jane Street

You had to solve a puzzle
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cxrupptedpat
228 posts
#4
Y by
i didnt get mines yet im on the west coast did anyone else not get it
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MC_ADe
167 posts
#5
Y by
i did but idt im getting it until like 2 years
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deduck
164 posts
#6
Y by
cxrupptedpat wrote:
i didnt get mines yet im on the west coast did anyone else not get it

i didnt get mine either and i filled out the puzzle form

they are scammers bruh
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MC413551
2228 posts
#7
Y by
xHypotenuse wrote:
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online

do you think it has something to do with the circle pattern?
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nxchman
21 posts
#8
Y by
Can you get the shirt if you qual for aime next year?
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Achelois
20 posts
#9
Y by
MC413551 wrote:
xHypotenuse wrote:
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online

do you think it has something to do with the circle pattern?

I was also thinking this! Maybe $r_i$ is related to producing those circles of varying radii? And I could see the $[0,1]^2$ having to do something with the square (if the shirt design is indeed produced by the humongous equation).
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EaZ_Shadow
1086 posts
#10 • 1 Y
Y by MathCosine
xHypotenuse wrote:
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online

I somehow got 2 of those jane street shirts :skull:
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LightningZ
190 posts
#11
Y by
I asked chatgpt for this and it's explanation was actually pretty helpful:
https://chatgpt.com/share/67c63f91-429c-8006-b341-5bc617d4c3d9
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Pengu14
395 posts
#12
Y by
I made ChatGPT write a program to calculate the expected value for the first 20 values of n with 10,000 trials each and this was the output:

Output
This post has been edited 3 times. Last edited by Pengu14, Mar 4, 2025, 12:07 AM
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Pengu14
395 posts
#13
Y by
This was the program:

Program
This post has been edited 1 time. Last edited by Pengu14, Mar 4, 2025, 12:07 AM
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alcumusftwgrind
47 posts
#14
Y by
i got the tshirt and i got a deck of cards too
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RedFireTruck
4206 posts
#15
Y by
based on @2above's outputs i think chatgpt solved it below

We begin by noting that for each \(x\in [0,1]^2\) the integrand is
\[
\left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\right) \bmod 2,
\]which is \(1\) if the number of disks covering \(x\) is odd and \(0\) if it is even. Therefore, the expected value of the integrand at a point \(x\) is the probability that \(x\) is covered by an odd number of disks.

Step 1. Coverage Probability for a Fixed Point

For a given ball \(B(X_i,r_i)\):
- The center \(X_i\) is uniformly distributed in \([0,1]^2\).
- The radius \(r_i\) is uniformly distributed on \(\left[0,\sqrt{\frac{3}{n\pi}}\right]\).

For a fixed \(x\) (assuming \(x\) is not too close to the boundary so that edge effects are negligible), the probability that \(x\) is covered by the \(i\)th disk is:
\[
\mathbb{P}\{x\in B(X_i,r_i)\} = \mathbb{E}[\pi r_i^2].
\]Since \(r_i\sim U\left(0,\sqrt{\frac{3}{n\pi}}\right)\), we have
\[
\mathbb{E}[r_i^2] = \frac{\left(\sqrt{\frac{3}{n\pi}}\right)^2}{3} = \frac{3/(n\pi)}{3} = \frac{1}{n\pi}.
\]Thus,
\[
\mathbb{E}[\pi r_i^2] = \pi\cdot\frac{1}{n\pi} = \frac{1}{n}.
\]
Step 2. Limiting Distribution of the Number of Coverings

Since each ball independently covers \(x\) with probability \(\frac{1}{n}\), the total number of coverings at \(x\) is a Binomial random variable:
\[
N_x \sim \operatorname{Binomial}\left(n, \frac{1}{n}\right).
\]As \(n\to\infty\), this Binomial distribution converges in distribution to a Poisson distribution with mean
\[
\lambda = n\cdot\frac{1}{n} = 1.
\]Thus,
\[
N_x \stackrel{d}{\to} \operatorname{Poisson}(1).
\]
Step 3. Computing the Probability of an Odd Number of Coverings

For a Poisson random variable \(N \sim \operatorname{Poisson}(1)\), the probability of having an odd number of events is:
\[
\mathbb{P}(N \text{ is odd}) = \frac{1 - e^{-2\cdot 1}}{2} = \frac{1 - e^{-2}}{2}.
\]This follows from the identity
\[
\mathbb{E}[(-1)^N] = e^{-2\lambda} \quad \text{and} \quad \mathbb{P}(N\text{ is odd}) = \frac{1 - \mathbb{E}[(-1)^N]}{2}.
\]
Step 4. Taking the Expectation over the Unit Square

Since the expected value of the integrand at any point \(x\) converges to \(\frac{1 - e^{-2}}{2}\) and the integration is over the unit square (area \(1\)), the expected value of the integral is simply:
\[
\mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x) \bmod 2\right) dx\right] \to \frac{1 - e^{-2}}{2}.
\]
Final Answer

Thus, the limit is:
\[
\lim_{n\to\infty} \mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\bmod 2\right)dx\right] = \frac{1 - e^{-2}}{2}.
\]
This post has been edited 1 time. Last edited by RedFireTruck, Mar 4, 2025, 3:10 AM
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Jaxman8
93 posts
#16
Y by
Yuh, I didn't get mine yet depsite filling out the form like 3 months ago. I was just wondering abt the t-shirt yesterday.
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maxamc
440 posts
#17
Y by
I did not get the email but got the link from some random place on AoPS and got the T-shirt using inspect to get the solve link.
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MathPerson12321
3606 posts
#18
Y by
I just got mine 2 days ago.
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derekwang2048
1189 posts
#19
Y by
i didnt get mine either :(
also on west coast
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xHypotenuse
732 posts
#20
Y by
EaZ_Shadow wrote:
xHypotenuse wrote:
Hello, for those who own the aime qual t-shirt for this year (it's dark blue with circles in the middle and this really weird, long equation), does anyone know what the equation is supposed to represent? Can't find it online

I somehow got 2 of those jane street shirts :skull:

Yeah same :skull:
Maybe it has something to do with me being enrolled in woot 1 (because I got 2 shirts as part of woot merch), are you too?
This post has been edited 1 time. Last edited by xHypotenuse, Mar 4, 2025, 7:21 AM
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xHypotenuse
732 posts
#21
Y by
RedFireTruck wrote:
based on @2above's outputs i think chatgpt solved it below

We begin by noting that for each \(x\in [0,1]^2\) the integrand is
\[
\left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\right) \bmod 2,
\]which is \(1\) if the number of disks covering \(x\) is odd and \(0\) if it is even. Therefore, the expected value of the integrand at a point \(x\) is the probability that \(x\) is covered by an odd number of disks.

Step 1. Coverage Probability for a Fixed Point

For a given ball \(B(X_i,r_i)\):
- The center \(X_i\) is uniformly distributed in \([0,1]^2\).
- The radius \(r_i\) is uniformly distributed on \(\left[0,\sqrt{\frac{3}{n\pi}}\right]\).

For a fixed \(x\) (assuming \(x\) is not too close to the boundary so that edge effects are negligible), the probability that \(x\) is covered by the \(i\)th disk is:
\[
\mathbb{P}\{x\in B(X_i,r_i)\} = \mathbb{E}[\pi r_i^2].
\]Since \(r_i\sim U\left(0,\sqrt{\frac{3}{n\pi}}\right)\), we have
\[
\mathbb{E}[r_i^2] = \frac{\left(\sqrt{\frac{3}{n\pi}}\right)^2}{3} = \frac{3/(n\pi)}{3} = \frac{1}{n\pi}.
\]Thus,
\[
\mathbb{E}[\pi r_i^2] = \pi\cdot\frac{1}{n\pi} = \frac{1}{n}.
\]
Step 2. Limiting Distribution of the Number of Coverings

Since each ball independently covers \(x\) with probability \(\frac{1}{n}\), the total number of coverings at \(x\) is a Binomial random variable:
\[
N_x \sim \operatorname{Binomial}\left(n, \frac{1}{n}\right).
\]As \(n\to\infty\), this Binomial distribution converges in distribution to a Poisson distribution with mean
\[
\lambda = n\cdot\frac{1}{n} = 1.
\]Thus,
\[
N_x \stackrel{d}{\to} \operatorname{Poisson}(1).
\]
Step 3. Computing the Probability of an Odd Number of Coverings

For a Poisson random variable \(N \sim \operatorname{Poisson}(1)\), the probability of having an odd number of events is:
\[
\mathbb{P}(N \text{ is odd}) = \frac{1 - e^{-2\cdot 1}}{2} = \frac{1 - e^{-2}}{2}.
\]This follows from the identity
\[
\mathbb{E}[(-1)^N] = e^{-2\lambda} \quad \text{and} \quad \mathbb{P}(N\text{ is odd}) = \frac{1 - \mathbb{E}[(-1)^N]}{2}.
\]
Step 4. Taking the Expectation over the Unit Square

Since the expected value of the integrand at any point \(x\) converges to \(\frac{1 - e^{-2}}{2}\) and the integration is over the unit square (area \(1\)), the expected value of the integral is simply:
\[
\mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x) \bmod 2\right) dx\right] \to \frac{1 - e^{-2}}{2}.
\]
Final Answer

Thus, the limit is:
\[
\lim_{n\to\infty} \mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\bmod 2\right)dx\right] = \frac{1 - e^{-2}}{2}.
\]

Interesting...thanks :)
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SirAppel
850 posts
#22
Y by
free shirt + deck of cards = fire
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Pengu14
395 posts
#23
Y by
RedFireTruck wrote:
based on @2above's outputs i think chatgpt solved it below

We begin by noting that for each \(x\in [0,1]^2\) the integrand is
\[
\left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\right) \bmod 2,
\]which is \(1\) if the number of disks covering \(x\) is odd and \(0\) if it is even. Therefore, the expected value of the integrand at a point \(x\) is the probability that \(x\) is covered by an odd number of disks.

Step 1. Coverage Probability for a Fixed Point

For a given ball \(B(X_i,r_i)\):
- The center \(X_i\) is uniformly distributed in \([0,1]^2\).
- The radius \(r_i\) is uniformly distributed on \(\left[0,\sqrt{\frac{3}{n\pi}}\right]\).

For a fixed \(x\) (assuming \(x\) is not too close to the boundary so that edge effects are negligible), the probability that \(x\) is covered by the \(i\)th disk is:
\[
\mathbb{P}\{x\in B(X_i,r_i)\} = \mathbb{E}[\pi r_i^2].
\]Since \(r_i\sim U\left(0,\sqrt{\frac{3}{n\pi}}\right)\), we have
\[
\mathbb{E}[r_i^2] = \frac{\left(\sqrt{\frac{3}{n\pi}}\right)^2}{3} = \frac{3/(n\pi)}{3} = \frac{1}{n\pi}.
\]Thus,
\[
\mathbb{E}[\pi r_i^2] = \pi\cdot\frac{1}{n\pi} = \frac{1}{n}.
\]
Step 2. Limiting Distribution of the Number of Coverings

Since each ball independently covers \(x\) with probability \(\frac{1}{n}\), the total number of coverings at \(x\) is a Binomial random variable:
\[
N_x \sim \operatorname{Binomial}\left(n, \frac{1}{n}\right).
\]As \(n\to\infty\), this Binomial distribution converges in distribution to a Poisson distribution with mean
\[
\lambda = n\cdot\frac{1}{n} = 1.
\]Thus,
\[
N_x \stackrel{d}{\to} \operatorname{Poisson}(1).
\]
Step 3. Computing the Probability of an Odd Number of Coverings

For a Poisson random variable \(N \sim \operatorname{Poisson}(1)\), the probability of having an odd number of events is:
\[
\mathbb{P}(N \text{ is odd}) = \frac{1 - e^{-2\cdot 1}}{2} = \frac{1 - e^{-2}}{2}.
\]This follows from the identity
\[
\mathbb{E}[(-1)^N] = e^{-2\lambda} \quad \text{and} \quad \mathbb{P}(N\text{ is odd}) = \frac{1 - \mathbb{E}[(-1)^N]}{2}.
\]
Step 4. Taking the Expectation over the Unit Square

Since the expected value of the integrand at any point \(x\) converges to \(\frac{1 - e^{-2}}{2}\) and the integration is over the unit square (area \(1\)), the expected value of the integral is simply:
\[
\mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x) \bmod 2\right) dx\right] \to \frac{1 - e^{-2}}{2}.
\]
Final Answer

Thus, the limit is:
\[
\lim_{n\to\infty} \mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\bmod 2\right)dx\right] = \frac{1 - e^{-2}}{2}.
\]

I’m not surprised e shows up in a probability question :)

ChatGPT is getting scary good though…
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derekwang2048
1189 posts
#24
Y by
EVERYONE MY T-SHIRT ARRIVED!!!!!!!!!!!
I've been waiting so long for this moment
and it did come with a deck of cards, but what the heck was that equation fr
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CharviA
33 posts
#25
Y by
derekwang2048 wrote:
EVERYONE MY T-SHIRT ARRIVED!!!!!!!!!!!
I've been waiting so long for this moment
and it did come with a deck of cards, but what the heck was that equation fr

the deck of cards is a puzzle if you look closely
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CharviA
33 posts
#26
Y by
I haven't solved it but you can get hints by scanning the QR code on the puzzle card
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LusenYao
27 posts
#27
Y by
CharviA wrote:
I haven't solved it but you can get hints by scanning the QR code on the puzzle card

where can I find the puzzle :blush:
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MC413551
2228 posts
#28
Y by
LusenYao wrote:
CharviA wrote:
I haven't solved it but you can get hints by scanning the QR code on the puzzle card

where can I find the puzzle :blush:

the puzzle is the deck of cards that they might've sent you
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LusenYao
27 posts
#30
Y by
alr thanks
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xHypotenuse
732 posts
#31
Y by
MC413551 wrote:
LusenYao wrote:
CharviA wrote:
I haven't solved it but you can get hints by scanning the QR code on the puzzle card

where can I find the puzzle :blush:

the puzzle is the deck of cards that they might've sent you

I think it was some "arrange the numbers in order" pyramid that took me an hour to solve...
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CharviA
33 posts
#32
Y by
xHypotenuse wrote:
MC413551 wrote:
LusenYao wrote:

where can I find the puzzle :blush:

the puzzle is the deck of cards that they might've sent you

I think it was some "arrange the numbers in order" pyramid that took me an hour to solve...

is that what it was? I was really confused the first time I tried to solve it.
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derekli
41 posts
#33
Y by
OMGOMGOMG HOW DO I GET IT
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derekwang2048
1189 posts
#34
Y by
by qualifying for aime and solving the puzzle
This post has been edited 1 time. Last edited by derekwang2048, Mar 12, 2025, 3:28 AM
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ChickensEatGrass
24 posts
#35
Y by
WHAT WAS THE PUZZLE'S ANSWER
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derekwang2048
1189 posts
#36
Y by
chill it was answer
This post has been edited 1 time. Last edited by derekwang2048, Mar 12, 2025, 3:31 AM
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Puck_0
263 posts
#38
Y by
RedFireTruck wrote:
based on @2above's outputs i think chatgpt solved it below

We begin by noting that for each \(x\in [0,1]^2\) the integrand is
\[
\left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\right) \bmod 2,
\]which is \(1\) if the number of disks covering \(x\) is odd and \(0\) if it is even. Therefore, the expected value of the integrand at a point \(x\) is the probability that \(x\) is covered by an odd number of disks.

Step 1. Coverage Probability for a Fixed Point

For a given ball \(B(X_i,r_i)\):
- The center \(X_i\) is uniformly distributed in \([0,1]^2\).
- The radius \(r_i\) is uniformly distributed on \(\left[0,\sqrt{\frac{3}{n\pi}}\right]\).

For a fixed \(x\) (assuming \(x\) is not too close to the boundary so that edge effects are negligible), the probability that \(x\) is covered by the \(i\)th disk is:
\[
\mathbb{P}\{x\in B(X_i,r_i)\} = \mathbb{E}[\pi r_i^2].
\]Since \(r_i\sim U\left(0,\sqrt{\frac{3}{n\pi}}\right)\), we have
\[
\mathbb{E}[r_i^2] = \frac{\left(\sqrt{\frac{3}{n\pi}}\right)^2}{3} = \frac{3/(n\pi)}{3} = \frac{1}{n\pi}.
\]Thus,
\[
\mathbb{E}[\pi r_i^2] = \pi\cdot\frac{1}{n\pi} = \frac{1}{n}.
\]
Step 2. Limiting Distribution of the Number of Coverings

Since each ball independently covers \(x\) with probability \(\frac{1}{n}\), the total number of coverings at \(x\) is a Binomial random variable:
\[
N_x \sim \operatorname{Binomial}\left(n, \frac{1}{n}\right).
\]As \(n\to\infty\), this Binomial distribution converges in distribution to a Poisson distribution with mean
\[
\lambda = n\cdot\frac{1}{n} = 1.
\]Thus,
\[
N_x \stackrel{d}{\to} \operatorname{Poisson}(1).
\]
Step 3. Computing the Probability of an Odd Number of Coverings

For a Poisson random variable \(N \sim \operatorname{Poisson}(1)\), the probability of having an odd number of events is:
\[
\mathbb{P}(N \text{ is odd}) = \frac{1 - e^{-2\cdot 1}}{2} = \frac{1 - e^{-2}}{2}.
\]This follows from the identity
\[
\mathbb{E}[(-1)^N] = e^{-2\lambda} \quad \text{and} \quad \mathbb{P}(N\text{ is odd}) = \frac{1 - \mathbb{E}[(-1)^N]}{2}.
\]
Step 4. Taking the Expectation over the Unit Square

Since the expected value of the integrand at any point \(x\) converges to \(\frac{1 - e^{-2}}{2}\) and the integration is over the unit square (area \(1\)), the expected value of the integral is simply:
\[
\mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x) \bmod 2\right) dx\right] \to \frac{1 - e^{-2}}{2}.
\]
Final Answer

Thus, the limit is:
\[
\lim_{n\to\infty} \mathbb{E}\left[\int_{[0,1]^2} \left(\sum_{i=1}^{n}1_{B(X_i,r_i)}(x)\bmod 2\right)dx\right] = \frac{1 - e^{-2}}{2}.
\]

I showed my dad the question and he got the same answer... can't say i even understand what the question is saying but if my dad got the answer and chat got the answer its prob correct
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RedFireTruck
4206 posts
#39
Y by
zamn ur dad mite be the goat ngl
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ChickensEatGrass
24 posts
#40
Y by
derekwang2048 wrote:
chill it was answer

Thank you so much; that answer helped me more than you could ever know. With your guidance, I was able to win MATHCOUNTS Nats before it even happened, make the IMO team, and achieve the Fields Award.
This post has been edited 1 time. Last edited by ChickensEatGrass, Thursday at 2:50 AM
Reason: r
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CharviA
33 posts
#41
Y by
where can I find the puzzle :blush:[/quote]

the puzzle is the deck of cards that they might've sent you[/quote]

I think it was some "arrange the numbers in order" pyramid that took me an hour to solve...[/quote]

this is the one you need to solve to get the cards not the one on the cards
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RainbowSquirrel53B
581 posts
#42
Y by
I got it like a week ago along with a green shirt, I took Woot and there was a form you could fill out
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