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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Apr 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
A Interesting Puzzle Again
Iwato   0
9 minutes ago
Source: One of my classmates, Zhu MingYu
Given n∈ N+, for all m≤n, proof that: At least one of the mininum points of φ(m) is staying in [n,2n]. For more, to check the bound value and if it's strict. By the way, to consider the situation under the range that m is bigger than n, and the total condition or the particular ones(about Euler's function's Value Distribution). At least, the Betrand-Chebyshv and Legendre's Conjecture(the latter idea is a improvement of the beyond one, which wassuspected by myself before I know this :D) are deserved to consider.
0 replies
Iwato
9 minutes ago
0 replies
PD is the angle bisector of <BPC
the_universe6626   3
N 18 minutes ago by ja.
Source: Janson MO 6 P1
Let $ABC$ be an acute triangle with $AB<AC$. The angle bisector of $\angle{BAC}$ intersects $BC$ at $D$, and the perpendicular to $AD$ at $D$ intersects $AB$ and $AC$ at $E, F$ respectively. Suppose $(AEF)$ and $(ABC)$ intersect again at $P\neq A$, prove that $PD$ is the angle bisector of $\angle{BPC}$.

(Proposed by quacksaysduck)
3 replies
the_universe6626
Feb 21, 2025
ja.
18 minutes ago
help me guyss
Bet667   1
N 25 minutes ago by Bet667
i want to learn functionl equation.Can you guys give me some advise to learn functional equations :starwars:
1 reply
Bet667
Yesterday at 3:49 PM
Bet667
25 minutes ago
Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $  a,b,c>0 $ and $a+ 2b+c =1.$ Prove that
$$\frac 1a + \frac 1{2b} + \frac 1c+abc \geq\frac{487}{54} $$Let $  a,b,c>0 $ and $2a+ b+2c = 1.$ Prove that
$$\frac 1a + \frac 2b + \frac 1c+abc \geq\frac{1945}{108} $$
2 replies
sqing
2 hours ago
sqing
an hour ago
Modular Arithmetic Handout
MathCosine   16
N 2 hours ago by MathCosine
Hi everyone,

I recently created a handout on modular arithmetic for a local math club. I thought it would help quite a lot with understanding basic properties, as modular arithmetic is a very popular intermediate step in number theory problems, so I decided to leave it here as a resource for anyone who needs it. Feel free to share it around, and hope it helps!

Sincerely,
MathCosine
16 replies
MathCosine
Apr 7, 2025
MathCosine
2 hours ago
idk12345678 Math Contest
idk12345678   0
2 hours ago
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications


*I mightve done them wrong feel free to ask about an answer
0 replies
idk12345678
2 hours ago
0 replies
How to get good at comp math
fossasor   9
N 3 hours ago by Inaaya
I'm a rising ninth grader who wasn't in the school math league this year, and basically put aside comp math for a year. Unfortunately, that means that now that I'm in high school and having the epiphany about how important comp math actually is, and how much it would help my chances of getting involved in other math-related programs. In addition, I do enjoy math in general, and suspect that things like the AMCs are probably going to be some of the best practice I can get. What this all means is that I'm trying to go from mediocre to orz, 2 years after I probably should have started if I wanted to be any good.

So my question is: how do I get good at comp math?

This year, my scores on AMC 10 (and these are the highest I've ever gotten) were a 73.5 and an 82.5 (AMC 8 was 21/25, but that doesn't matter much). This is not good enough to qualify for AIME, and I probably need to raise my performance on each by at least 10 points. I've been decently good in the past at Number Theory, but I need to work on Geo and Combinatorics, and I'm trying to find the best resources to do that. My biggest flaw is probably not knowing many algorithms like Stars and Bars, and the path is clear here (learn them) but I'm still not sure which ones I need to know.

I'm aware that some of this advice is going to be something like "Practice 5 hours a day and start hardgrinding" or something along those lines. Unfortunately, I have other extracurriculars I need to balance, and for me, time is a limiting resource. My parents are somewhat frowning upon me doing a lot of comp math, which limits my time as well. I have neither the time nor motivation to do more than an hour a day, and in practice, I don't think I can be doing that consistently. As such, I would need to make that time count.

I know this is a very general question, and that aops is chock-full of detailed advice for math competitions. However, I'd appreciate it if anyone here could help me out, or show me the best resources I should use to get started. What mocks are any good, or what textbooks should I use? Where do I get the best practice with the shortest time? Is there some place I can find a list of useful formulas that have appeared in math comps before?

All advice is welcome!

9 replies
fossasor
5 hours ago
Inaaya
3 hours ago
1990 AMC 12 #24
dft   17
N 3 hours ago by Bread10
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
\[ \begin{tabular}{c c c c} 
{} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker}  \\
\textbf{Boys:} & 71 & 81 & 79   \\
\textbf{Girls:} & 76 & 90 & ?   \\
\textbf{Boys and Girls:} & 74 & 84 &   \\
\end{tabular}
 \]
$ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 84 \quad\textbf{(E)}\ 85 $
17 replies
dft
Dec 31, 2011
Bread10
3 hours ago
2025 Math and AI 4 Girls Competition: Win Up To $1,000!!!
audio-on   35
N 4 hours ago by Lightybird
Join the 2025 Math and AI 4 Girls Competition for a chance to win up to $1,000!

Hey Everyone, I'm pleased to announce the dates for the 2025 MA4G Competition are set!
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (@ 11:59pm PST).

Applicants will have one month to fill out an application with prizes for the top 50 contestants & cash prizes for the top 20 contestants (including $1,000 for the winner!). More details below!

Eligibility:
The competition is free to enter, and open to middle school female students living in the US (5th-8th grade).
Award recipients are selected based on their aptitude, activities and aspirations in STEM.

Event dates:
Applications will open on March 22nd, 2025, and they will close on April 26th, 2025 (by 11:59pm PST)
Winners will be announced on June 28, 2025 during an online award ceremony.

Application requirements:
Complete a 12 question problem set on math and computer science/AI related topics
Write 2 short essays

Prizes:
1st place: $1,000 Cash prize
2nd place: $500 Cash prize
3rd place: $300 Cash prize
4th-10th: $100 Cash prize each
11th-20th: $50 Cash prize each
Top 50 contestants: Over $50 worth of gadgets and stationary


Many thanks to our current and past sponsors and partners: Hudson River Trading, MATHCOUNTS, Hewlett Packard Enterprise, Automation Anywhere, JP Morgan Chase, D.E. Shaw, and AI4ALL.

Math and AI 4 Girls is a nonprofit organization aiming to encourage young girls to develop an interest in math and AI by taking part in STEM competitions and activities at an early age. The organization will be hosting an inaugural Math and AI 4 Girls competition to identify talent and encourage long-term planning of academic and career goals in STEM.

Contact:
mathandAI4girls@yahoo.com

For more information on the competition:
https://www.mathandai4girls.org/math-and-ai-4-girls-competition

More information on how to register will be posted on the website. If you have any questions, please ask here!


35 replies
audio-on
Jan 26, 2025
Lightybird
4 hours ago
amc 10 cooked
PaperMath   118
N 5 hours ago by parnikap
Guys any tips to staying calm? (please don't reply with take a deep breath)
118 replies
1 viewing
PaperMath
Nov 5, 2024
parnikap
5 hours ago
9 Practice AIME Exam
Melissa.   17
N 5 hours ago by fake123
(This practice test is designed to be slightly harder than the real test. I would recommend you take this like a real test, using a 3 hour time limit and no calculator.)

Let me know any suggestions for improvement on test quality, difficulty, problem selection, problem placement, test topics, etc. for the next tests that I make!

Practice AIME

1.
Positive integers a, b, and c satisfy a + b + c = 49 and ab + bc + ca = 471. Find the value of the product abc.

2.
Find the integer closest to the value of (69^(1/2) + 420^(1/2))^2.

3.
Let G and A be two points that are 243 units apart. Suppose A_1 is at G, and for n > 1, A_n is the point on line GA such that A_nA_(n-1) = 243, and A_n is farther from A than G. Let L be the locus of points T such that GT + A_6T = 2025. Find the maximum possible distance from T to line GA as T varies across L.

4.
Find the value of (69 + 12 * 33^(1/2))^(1/2) + (69 - 12 * 33^(1/2))^(1/2).

5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.

6.
Find the limit of (1x^2 + 345x^6)/(5x^6 + 78x + 90) as x approaches infinity.

7.
Find the slope of the line tangent to the graph of y = 6x^2 + 9x + 420 at the point where y = 615 and x is positive.

8.
Find the smallest positive integer n such that the sum of the positive integer divisors of n is 1344.

9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.

10.
Let n be the smallest positive integer in base 10 such that the base 2 expression of 60n contains an odd number of 1’s. Find the sum of the squares of the digits of n.

11.
Find the sum of the 7 smallest positive integers n such that n is a multiple of 7, and the repeating decimal expansion of 1/n does not have a period of 6.

12.
Let n be an integer from 1 to 999, inclusive. How many different numerators are possible when n/1000 is written as a common fraction in lowest terms?

13.
How many ways are there to divide a pile of 15 indistinguishable bricks?

14.
Let n be the unique 3-digit positive integer such that the value of the product 100n can be expressed in bases b, b + 1, b + 2, and b + 3 using only 0’s and 1’s, for some integer b > 1. Find n.

15.
For positive integers n, let f(n) be the sum of the positive integer divisors of n. Suppose a positive integer k is untouchable if there does not exist a positive integer a such that f(a) = k + a. For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.

Answer key:
WARNING: SPOILERS!!!
17 replies
Melissa.
Apr 8, 2025
fake123
5 hours ago
9 Will I make AMO?
imagien_bad   7
N Yesterday at 10:50 PM by alcumusftwgrind
Hi everyone, I got a 100.5 on AMC 12A 2024 what is my chance to make USAMO 2025? (i did not do 12B btw)
7 replies
imagien_bad
Nov 23, 2024
alcumusftwgrind
Yesterday at 10:50 PM
Rutgers Expo in Problem Solving 2025 by OMMC
DottedCaculator   2
N Yesterday at 10:32 PM by Inaaya
Hello to all creative problem solvers,

Do you want a life changing math experience?
Do you want to see me in real life?

Check out the
Rutgers Expo in Problem Solving (REPS) by OMMC!

What is OMMC?

OMMC is presenting to you its next major event: in-person this time! This spring, OMMC is hosting its THIRD IN-PERSON event, where we will be presenting various speakers of math, holding breakout sessions, games and friendly competitions, and providing a math hub for people all over to learn and enjoy. No math experience is needed, and elementary, middle and high schoolers can all register!

The Rutgers Expo in Problem Solving will take place on Saturday, April 19th, 2025, 1 PM Eastern Time.
The venue is the Science Engineering Complex at Rutgers New-Brunswick. This is 96 Frelinghuysen Rd, Piscataway, NJ 08854.


Event includes:

[list]
[*]Math speakers, including university professors
[*]Activities including estimathon and mini math competition WITH PRIZES
[*]Itinerary coming soon!
[/list]

This event is completely FREE to all students!
Fill out the registration form linked below to sign-up for this event and answer some important questions.
https://docs.google.com/forms/d/e/1FAIpQLSeAr2Nul9MBO_adVzHN9Rsrc8yQEjzxPXZHZ-LUFf-zWcwR7A/viewform?usp=preview


Students from anywhere can attend, as long as you can commute to the venue. Email us at ommcofficial@gmail.com for any questions or concerns.

We can’t wait to see you there!
- REPS Team

OMMC’S 2025 EVENTS ARE SPONSORED BY:

[list]
[*]Nontrivial Fellowship
[*]Citadel
[*]Jane Street
[/list]

2 replies
DottedCaculator
Yesterday at 8:46 PM
Inaaya
Yesterday at 10:32 PM
Intro/intermediate books in terms of importance
Aaron_Q   6
N Yesterday at 9:20 PM by Aaron_Q
Hey everyone,
what are your opinions on the most critical (NEED to learn) intro/intermediate books?
i might be cooked
6 replies
Aaron_Q
Tuesday at 11:19 PM
Aaron_Q
Yesterday at 9:20 PM
Hard number theory
Hip1zzzil   13
N Mar 30, 2025 by Hip1zzzil
Source: FKMO 2025 P6
Positive integers $a, b$ satisfy both of the following conditions.
For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$.
There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$.
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.
13 replies
Hip1zzzil
Mar 30, 2025
Hip1zzzil
Mar 30, 2025
Hard number theory
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G H BBookmark kLocked kLocked NReply
Source: FKMO 2025 P6
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Hip1zzzil
14 posts
#1
Y by
Positive integers $a, b$ satisfy both of the following conditions.
For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$.
There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$.
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.
This post has been edited 4 times. Last edited by Hip1zzzil, Mar 30, 2025, 1:07 PM
Reason: Better
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Acorn-SJ
59 posts
#2 • 1 Y
Y by seoneo
Please copy the original wording, it’s mildly annoying to watch some phrases missing.
In particular, $m$ is supposed to be a positive integer.

EDIT: $n$ is an integer not a positive integer
This post has been edited 1 time. Last edited by Acorn-SJ, Mar 30, 2025, 11:25 AM
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pokmui9909
185 posts
#3 • 1 Y
Y by seoneo
This would be better:
Quote:
Positive integers $a, b$ satisfy both of the following conditions.
  • For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$.
  • There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$.
Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.
This post has been edited 2 times. Last edited by pokmui9909, Mar 30, 2025, 7:10 AM
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whwlqkd
89 posts
#4
Y by
Anyone solved it?
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Acorn-SJ
59 posts
#5
Y by
a failed attempt
This post has been edited 1 time. Last edited by Acorn-SJ, Mar 30, 2025, 6:20 AM
Reason: Typo
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Lufin
9 posts
#6 • 2 Y
Y by Acorn-SJ, jjkim0336
In the original paper, n is an integer, and does not have to be a positive integer.
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seoneo
339 posts
#7
Y by
(Sketch)

From the condition, $a$ and $b$ are square free and coprime.
By the infinite descent method, we have relatively prime $a \alpha, b \beta, \zeta, \omega$ so that $a \alpha^2 +b \beta^2 = \zeta^2 + \omega^2$.

Now consider the expression
\[ (\zeta t + p)^2 + (\omega t +q)^2 -a(\alpha t +r)^2 -b(\beta t +s)^2 = 2(\zeta p + \omega q -a\alpha r -b \beta s)t + p^2 +q^2 -ar^2 -bs^2 \]From Bezout, we have $p,q,r,s$ such that
\[ \zeta p + \omega q -a\alpha r -b \beta s =1 \]so we have all ever, or all odd $n$.

By changing parity of $p^2 +q^2 -ar^2 -bs^2 $, we have all integers.

PS. I thought switching the parity would be easy, but it's actually more subtle than I first thought.
This post has been edited 1 time. Last edited by seoneo, Mar 30, 2025, 8:17 AM
Reason: To add PS
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seoneo
339 posts
#8
Y by
This FKMO problem was not accurately translated at first. If the original wording is not preserved in the current post, it may be better to create a properly translated version and redirect others to that post.

P.S. It looks like it's been fixed now.
This post has been edited 1 time. Last edited by seoneo, Mar 31, 2025, 6:31 AM
Reason: To fix grammer.
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segment
24 posts
#10 • 1 Y
Y by MihaiT
seoneo wrote:
(Sketch)

From the condition, $a$ and $b$ are square free and coprime.
By the infinite descent method, we have relatively prime $a \alpha, b \beta, \zeta, \omega$ so that $a \alpha^2 +b \beta^2 = \zeta^2 + \omega^2$.

Now consider the expression
\[ (\zeta t + p)^2 + (\omega t +q)^2 -a(\alpha t +r)^2 -b(\beta t +s)^2 = 2(\zeta p + \omega q -a\alpha r -b \beta s)t + p^2 +q^2 -ar^2 -bs^2 \]From Bezout, we have $p,q,r,s$ such that
\[ \zeta p + \omega q -a\alpha r -b \beta s =1 \]so we have all ever, or all odd $n$.

By changing parity of $p^2 +q^2 -ar^2 -bs^2 $, we have all integers.

Solved same, the motivation was experimenting $a=b=1$, realizing that we can make a linear function of one variable.
I thought \[ \zeta p + \omega q -a\alpha r -b \beta s =1 \]is not enough for the case when $a,b,\alpha,\beta,\gamma,\delta$ are all odd, so for that case I used \[ \zeta p + \omega q -a\alpha r -b \beta s =2 \]and eventually I got all odds, $4k$s, $4k+2$s.
This post has been edited 2 times. Last edited by segment, Mar 30, 2025, 8:06 AM
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MihaiT
748 posts
#11
Y by
very nice!
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GreekIdiot
167 posts
#12
Y by
Acorn-SJ wrote:
Please copy the original wording, it’s mildly annoying to watch some phrases missing.
In particular, $m$ is supposed to be a positive integer.

Doesnt change a lot if $m$ is negative now, does it?
Also in case $ab$ is not squarefree then $m=1$ is a positive integer...
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Acorn-SJ
59 posts
#13
Y by
GreekIdiot wrote:
Acorn-SJ wrote:
Please copy the original wording, it’s mildly annoying to watch some phrases missing.
In particular, $m$ is supposed to be a positive integer.

Doesnt change a lot if $m$ is negative now, does it?
Also in case $ab$ is not squarefree then $m=1$ is a positive integer...


Yes, you could say that, but better preserve the wording for the sake of archiving. Also, a much more important phrase was omitted, so I added that in.
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GreekIdiot
167 posts
#14
Y by
yeah, just saw the edit
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Hip1zzzil
14 posts
#15
Y by
Thank you for all the replies, Sorry for my bad English. I'm working on it though :)
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