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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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Introduction to Algebra A
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Introduction to Algebra B Self-Paced

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Intermediate Algebra
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Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
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MATHCOUNTS/AMC 8 Advanced
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Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
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Programming

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Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 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
number theory problem
danilorj   0
3 hours ago
Let $t$ be an integer, show that there are infinite perfect squares of the form $3t^2+4t+5$
0 replies
danilorj
3 hours ago
0 replies
D1040 : A general and strange result
Dattier   0
4 hours ago
Source: les dattes à Dattier
Let $f \in C([0,1];[0,1])$ bijective, $f(0)=0$ and $(a_k) \in [0,1]^\mathbb N$ with $ \sum \limits_{k=0}^{+\infty} a_k$ converge.

Is it true that $\sum \limits_{k=0}^{+\infty} \sqrt{f(a_k)\times f^{-1}(a_k)}$ converge?
0 replies
Dattier
4 hours ago
0 replies
Trigo or Complex no.?
hzbrl   6
N 5 hours ago by GreenKeeper
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
6 replies
hzbrl
May 27, 2025
GreenKeeper
5 hours ago
Handouts/Resources on Limits.
Saucepan_man02   0
Today at 3:54 AM
Could anyone kindly share some resources/handouts on limits?
0 replies
Saucepan_man02
Today at 3:54 AM
0 replies
EGMO (geo) Radical Center Question
gulab_jamun   9
N Today at 12:58 AM by MathRook7817
For this theorem, Evan says that the power of point $P$ with respect to $\omega_1$ is greater than 0 if $P$ lies between $A$ and $B$. (I've underlined it). But, I'm a little confused as I thought the power was $OP^2 - r^2$ and since $P$ is inside the circle, wouldn't the power be negative since $OP < r$?
9 replies
gulab_jamun
May 25, 2025
MathRook7817
Today at 12:58 AM
9 point circle?!?!??!?
Maximilian113   32
N Today at 12:47 AM by NicoN9
Source: 2025 AIME II P5
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees.

IMAGE
32 replies
Maximilian113
Feb 13, 2025
NicoN9
Today at 12:47 AM
Close to JMO, but not close enough
isache   6
N Today at 12:03 AM by mathprodigy2011
Im currently a freshman in hs, and i rlly wanna make jmo in sophmore yr. Ive been cooking at in-person competitions recently (ucsd hmc, scmc, smt, mathcounts) but I keep fumbling jmo. this yr i had a 133.5 on 10b and a 9 on aime. How do i get that up by 20 points to a 240?
6 replies
isache
May 28, 2025
mathprodigy2011
Today at 12:03 AM
Frustration with Olympiad Geo
gulab_jamun   13
N Today at 12:03 AM by mathprodigy2011
Ok, so right now, I am doing the EGMO book by Evan Chen, but when it comes to problems, there are some that just genuinely frustrate me and I don't know how to deal with them. For example, I've spent 1.5 hrs on the second to last question in chapter 2, and used all the hints, and I still am stuck. It just frustrates me incredibly. Any tips on managing this? (or.... am I js crashing out too much?)
13 replies
gulab_jamun
May 29, 2025
mathprodigy2011
Today at 12:03 AM
Large grid
kevinmathz   13
N Yesterday at 6:48 PM by StressedPineapple
Source: 2020 AOIME #12
Let $m$ and $n$ be odd integers greater than $1.$ An $m\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099.$
13 replies
kevinmathz
Jun 7, 2020
StressedPineapple
Yesterday at 6:48 PM
Another Cubic Curve!
v_Enhance   165
N Yesterday at 5:55 PM by maromex
Source: USAMO 2015 Problem 1, JMO Problem 2
Solve in integers the equation
\[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]
165 replies
v_Enhance
Apr 28, 2015
maromex
Yesterday at 5:55 PM
Projections and Tangents
franchester   43
N Yesterday at 4:56 PM by StressedPineapple
Source: 2020 AOIME Problem 15
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$.
43 replies
franchester
Jun 7, 2020
StressedPineapple
Yesterday at 4:56 PM
Registrations Open for IOQM Level Up Test 2025!
oly01230   0
Yesterday at 5:13 AM
Registrations Open for IOQM Level Up Test 2025!

Hello everyone!

Are you a middle or high school student passionate about math competitions like IOQM, NMTC, RMO, and beyond? Do you want to benchmark your preparation, identify your strengths, and get mentored by some of the best minds in Olympiad math?

Then you should definitely check out the IOQM Level Up Test, organized by Narayana Prodigy – a platform dedicated to identifying and mentoring the top young math talent in the country.

- Register here: https://ioqm.co.in/

What is the IOQM Level Up Test?
IOQM Level Up is a nationwide simulation test built on the latest IOQM trends. It is designed for students of Classes 8 to 12 who aim to crack the IOQM, qualify for RMO, and ultimately represent India at the IMO. This is more than just a test – it’s a gateway to deeper mentorship, curated resources, and Prodigy workshops.

Why You Should Participate
- Real IOQM-level experience: Designed by Olympiad experts and past IOQM/RMO mentors
- Detailed test analysis: Understand your strengths and areas of improvement
- Top scorer mentorship: Shortlisted students get access to advanced training sessions
- Access to learning resources: Problem sets, video solutions, and more
- Stand a chance to join the elite Narayana Prodigy program

Test Details
- Test Date: 22 June 2025
- Duration: 3 Hours
- Mode: Online & Offline (select Narayana Centres)
- For: Students of Classes 8 to 12 aspiring for IOQM 2025
- Fees: Rs. 50 for 4 Mock Tests complete with SWOT analysis and National benchmarking Report

- If you're already preparing for Olympiads, this is the test you shouldn’t miss!

About Narayana Prodigy
Narayana Prodigy is the Olympiad division of Narayana Group, known for training some of the brightest students for JEE, NEET, and Science/Math Olympiads. We believe in nurturing academic prodigies from an early stage through workshops, mentoring, and customized learning paths.

Want to Know More?
Explore more about our Olympiad programs and IOQM mentoring initiative:
- https://prodigy.narayanagroup.com
- For queries: support.prodigy@narayanagroup.com

Take the first step toward your Olympiad journey. Register now and Level Up!
0 replies
oly01230
Yesterday at 5:13 AM
0 replies
Perfect Square Dice
asp211   69
N Yesterday at 4:55 AM by ohiorizzler1434
Source: 2019 AIME II #4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
69 replies
asp211
Mar 22, 2019
ohiorizzler1434
Yesterday at 4:55 AM
Mustang Math Recruitment is Open!
MustangMathTournament   7
N Yesterday at 4:54 AM by ohiorizzler1434
The Interest Form for joining Mustang Math is open!

Hello all!

We're Mustang Math, and we are currently recruiting for the 2025-2026 year! If you are a high school or college student and are passionate about promoting an interest in competition math to younger students, you should strongly consider filling out the following form: https://link.mustangmath.com/join. Every member in MM truly has the potential to make a huge impact, no matter your experience!

About Mustang Math

Mustang Math is a nonprofit organization of high school and college volunteers that is dedicated to providing middle schoolers access to challenging, interesting, fun, and collaborative math competitions and resources. Having reached over 4000 U.S. competitors and 1150 international competitors in our first six years, we are excited to expand our team to offer our events to even more mathematically inclined students.

PROJECTS
We have worked on various math-related projects. Our annual team math competition, Mustang Math Tournament (MMT) recently ran. We hosted 8 in-person competitions based in Washington, NorCal, SoCal, Illinois, Georgia, Massachusetts, Nevada and New Jersey, as well as an online competition run nationally. In total, we had almost 900 competitors, and the students had glowing reviews of the event. MMT International will once again be running later in August, and with it, we anticipate our contest to reach over a thousand students.

In our classes, we teach students math in fun and engaging math lessons and help them discover the beauty of mathematics. Our aspiring tech team is working on a variety of unique projects like our website and custom test platform. We also have a newsletter, which, combined with our social media presence, helps to keep the mathematics community engaged with cool puzzles, tidbits, and information about the math world! Our design team ensures all our merch and material is aesthetically pleasing.

Some highlights of this past year include 1000+ students in our classes, AMC10 mock with 150+ participants, our monthly newsletter to a subscriber base of 6000+, creating 8 designs for 800 pieces of physical merchandise, as well as improving our custom website (mustangmath.com, 20k visits) and test-taking platform (comp.mt, 6500+ users).

Why Join Mustang Math?

As a non-profit organization on the rise, there are numerous opportunities for volunteers to share ideas and suggest projects that they are interested in. Through our organizational structure, members who are committed have the opportunity to become a part of the leadership team. Overall, working in the Mustang Math team is both a fun and fulfilling experience where volunteers are able to pursue their passion all while learning how to take initiative and work with peers. We welcome everyone interested in joining!

More Information

To learn more, visit https://link.mustangmath.com/RecruitmentInfo. If you have any questions or concerns, please email us at contact@mustangmath.com.

https://link.mustangmath.com/join
7 replies
MustangMathTournament
May 24, 2025
ohiorizzler1434
Yesterday at 4:54 AM
Putnam 2003 A2
btilm305   9
N Apr 16, 2025 by KAME06
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]
9 replies
btilm305
Jun 22, 2011
KAME06
Apr 16, 2025
Putnam 2003 A2
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btilm305
439 posts
#1 • 1 Y
Y by Adventure10
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]
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Kent Merryfield
18574 posts
#2 • 5 Y
Y by mathman523, vsamc, GoodPerson, Adventure10, Sagnik123Biswas
More from my old document:

To start with, if any of these numbers is zero, one of the quantities on the left hand side is zero and the right hand side is clearly larger than the other side. So assume that all of the $a$s and $b$s are positive. For each $i, 1 \le i \le n,$ let $s_i = a_i + b_i.$ Let $\alpha_i = \frac{a_i}{s_i}$ and $\beta_i = \frac{b_i}{s_i} .$ Note that $\alpha_i + \beta_i = 1$ for all $i.$ Divide both sides of the inequality by $(s_1s_2\cdots s_n)^{1/n}.$ The result is that our inequality is equivalent to $(\alpha_1\alpha_2\cdots\alpha_n)^{1/n} + (\beta_1\beta_2\cdots\beta_n)^{1/n} \le 1.$ The arithmetic mean-geometric mean inequality (AM-GM inequality) now shows that $(\alpha_1\alpha_2\cdots\alpha_n)^{1/n} \le  \frac{\alpha_1 + \alpha_2 + \cdots + \alpha_n}n$ and that $(\beta_1\beta_2\cdots\beta_n)^{1/n} \le\frac{\beta_1 + \beta_2 + \cdots + \beta_n}n .$ So,
$(\alpha_1\alpha_2\cdots\alpha_n)^{1/n} + (\beta_1\beta_2\cdots\beta_n)^{1/n} \le  \frac{\alpha_1 + \beta_1 + \alpha_2 + \beta_2 + \cdots + \alpha_n + \beta_n}n$ $=  \frac{n}{n}  = 1$ and we are done.
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prasanna1712
109 posts
#3 • 2 Y
Y by Adventure10, Mango247
It seems trivial using Holder's extended inequality

see http://www.artofproblemsolving.com/Resources/Papers/MildorfInequalities.pdf
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mathbuzz
803 posts
#4 • 1 Y
Y by Adventure10
if any of $a_1,...,a_n,b_1,....b_n$ is zero , then the inequality is trivial.so , consider the case that all of them are positive.
let $a_i=x_i^n$ , $b_i=y_i^n$ .math=inline]$x_1,...,x_n ,y_1,...,y_n$[/math] are all [math=inline]$>0$[/math
then the inequality is equivalent to showing that,
$\prod{[x_i^n+y_i^n]} \ge (x_1x_2...x_n+y_1y_2....y_n)^n$
now , we use the following lemma--
Lemma-- $(p_1^n+1)......(p_n^n+1) \ge (p_1p_2....p_n+1)^n$ where all $p_i$'s are positive.
proof of the lemma--
let $p_1p_2...p_n+1=t$.
then by weighted AM-GM ,
we get $p_i^n+1=\frac{p_i^n}{t-1}.(t-1)+1 \ge t[\frac{p_i^{n(t-1)}}{(t-1)^{t-1}}]^{1/t}$
and the rest is trivial.
end of proof of lemma
now putting $p_i=\frac{x_i}{y_i}$ for all $i=1,2,..,n$ and we get the desired inequality :D :lol:
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sqing
42527 posts
#5 • 1 Y
Y by Adventure10
btilm305 wrote:
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]
2019 CGMO P6:
Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+
\sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$
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djmathman
7938 posts
#6
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An interesting fact about this problem: this is a special case of the Brunn-Minkowski Inequality.
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twbnrftw
248 posts
#7
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If any of $a_i=0,$ the inequality trivially holds. Similarly for $b_i.$

Hence considering $a_i,b_i>0,$ we will employ $\text{AM-GM}.$ inequality,

Clearly $\frac{\frac{a_1}{a_1+b_1}+\frac{a_2}{a_2+b_2}+\cdots+\frac{a_n}{a_n+b_n}}{n}\le\left(\frac{a_1}{a_1+b_1}\cdot\frac{a_2}{a_2+b_2}\cdots\frac{a_n}{a_n+b_n}\right)^{1/n}$


and, $\frac{\frac{b_1}{a_1+b_1}+\frac{b_2}{a_2+b_2}+\cdots+\frac{b_n}{a_n+b_n}}{n}\le\left(\frac{b_1}{a_1+b_1}\cdot\frac{b_2}{a_2+b_2}\cdots\frac{b_n}{a_n+b_n}\right)^{1/n}.$

Adding both of them we get $1\le\left(\frac{(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n}}{(a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}}\right).$

And hence this along with the trivial case gives us the desired inequality.$\quad\blacksquare$
This post has been edited 1 time. Last edited by twbnrftw, Jan 15, 2024, 4:07 AM
Reason: typo
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Sagnik123Biswas
421 posts
#8
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It suffices to execute a set of reversible steps. First let us raise both sides to the $n^{th}$ power. We obtain the following: $\sum_{k=0}^{n}\binom{n}{k}(a_1a_2a_3 \dots a_n)^{\frac{k}{n}}(b_1b_2b_3 \dots b_n)^{\frac{n-k}{n}} \leq (a_1 + b_1)(a_2+b_2)(a_3+b_3) \dots (a_n + b_n)$.

Now let us define $S_k$ to be the part of the expansion of $(a_1 + b_1)(a_2+b_2)(a_3+b_3) \dots (a_n + b_n)$ which contains $k$ terms of the form "$a_i$" and $n-k$ terms of the form "$b_i$". It then follows that $(a_1 + b_1)(a_2+b_2)(a_3+b_3) \dots (a_n + b_n) = \sum_{k=0}^{n}S_k$. So it suffices to show that $S_k \geq \binom{n}{k}(a_1a_2a_3 \dots a_n)^{\frac{k}{n}}(b_1b_2b_3 \dots b_n)^{\frac{n-k}{n}}$.


Next, we define $P_k$ to be the product of all terms in the expansion of $(a_1 + b_1)(a_2+b_2)(a_3+b_3) \dots (a_n + b_n)$ which contain $k$ terms of the form "$a_i$" and $n-k$ terms of the form "$b_i$". By the $AM-GM$ inequality, we know that $S_k \geq \binom{n}{k}(P_k)^{\frac{1}{\binom{n}{k}}}$.

If we are able to show that $\binom{n}{k}(P_k)^{\frac{1}{\binom{n}{k}}} = \binom{n}{k}(a_1a_2a_3 \dots a_n)^{\frac{k}{n}}(b_1b_2b_3 \dots b_n)^{\frac{n-k}{n}}$, then we should be done. Observe that $(P_k)^{\frac{1}{\binom{n}{k}}} =  (a_1a_2a_3 \dots a_n) ^ { \frac{\binom{n-1}{k-1}}{\binom{n}{k}}}  (b_1b_2b_3 \dots b_n) ^ { \frac{\binom{n-1}{k}}{\binom{n}{k}}}$. The reason this holds is that each $a_i$ appears in $\binom{n-1}{k-1}$ different terms of the expansion and likewise each $b_i$ appears in $\binom{n-1}{k}$ terms of the expansion.


Simplifying $P_k$, we see that $\frac{\binom{n-1}{k-1}}{\binom{n}{k}} = \frac{(n-1)!}{(n-k)!(k-1)!} \times \frac{k!(n-k)!}{n!} =  \frac{k}{n}$. By Pascal's Identity, it must follow that $\frac{\binom{n-1}{k}}{\binom{n}{k}} = \frac{\binom{n}{k} - \binom{n-1}{k-1}}{\binom{n}{k}} = \frac{n-k}{n}$. Thus, it is indeed true that $\binom{n}{k}(P_k)^{\frac{1}{\binom{n}{k}}} = \binom{n}{k}(a_1a_2a_3 \dots a_n)^{\frac{k}{n}}(b_1b_2b_3 \dots b_n)^{\frac{n-k}{n}}$, thereby completing the proof.
This post has been edited 1 time. Last edited by Sagnik123Biswas, Jan 23, 2024, 1:34 PM
Reason: Typo
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sanyalarnab
947 posts
#9 • 1 Y
Y by mqoi_KOLA
Easy but cute problem :love:
By AM GM Inequality,
$$\frac{1}{n} \sum_{k=1}^n \frac{a_k}{a_k+b_k} \geq \left( \prod_{k=1}^n \frac{a_k}{a_k+b_k}\right)^{1/n}$$$$\frac{1}{n} \sum_{k=1}^n \frac{b_k}{a_k+b_k} \geq \left( \prod_{k=1}^n \frac{b_k}{a_k+b_k}\right)^{1/n}$$Now adding these two inequalities, we have
$$1=\frac{1}{n} \sum_{k=1}^n 1 \geq \left( \prod_{k=1}^n \frac{a_k}{a_k+b_k}\right)^{1/n}+\left( \prod_{k=1}^n \frac{b_k}{a_k+b_k}\right)^{1/n}$$This implies the statement to be proved. $\blacksquare$
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KAME06
161 posts
#10
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WLOG, if $a_1=0$, then we must prove $(b_1b_2...b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n} \Leftrightarrow b_1b_2...b_n \le (a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n)$, but $LHS$ is $RHS$'s addend, so the inequality is true
If they are positive reals:
Let $A_k$ the set of all possible products, with $n$ factors, made by $k$ distinct elements from $\{a_1, a_2, ..., a_n \}$ and $n-k$ distinct elements from $\{b_1, b_2, ..., b_n \}$ such there aren't $a, b$ with the same index. There are $\binom{n}{k}$ ways to select $k$ distinct elements from $\{a_1, a_2, ..., a_n \}$. We can't select same indeces and the product has $n$ factors, so there is $1$ way to select $n-k$ distinct elements from $\{b_1, b_2, ..., b_n \}$. All of that implies $|A_k|=\binom{n}{k}$.
Then using MA-MG: $\sum_{x \in A_k} x \ge \binom{n}{k} \sqrt[n]{\left (\prod_{i=1}^{n}a_i \right)^k \left(\prod_{i=1}^{n}b_i \right )^{n-k}}$
$\forall k=1, 2, ..., n$
Now, consider the product $(a_1+b_1)(a_2+b_2)...(a_n+b_n)$ notice that each addend is made by $k$ distinct $a$'s and $n-k$ distinct $b$'s, $\forall k=1, 2, ..., n$. Also, each element of $A_k$ is an addend of $(a_1+b_1)(a_2+b_2)...(a_n+b_n)$. That implies that $(a_1+b_1)(a_2+b_2)...(a_n+b_n)=\sum_{k=0}^n \sum_{x \in A_k}x$.
We conclude that, adding all the inequalities we obtained using MA-MG, for $k=0, 1, 2, ..., n$:
$$\sum_{k=0}^n \binom{n}{k} \sqrt[n]{\left (\prod_{i=1}^{n}a_i \right)^k \left(\prod_{i=1}^{n}b_i \right )^{n-k}} \le \sum_{k=0}^n \sum_{x \in A_k}x$$$$\Leftrightarrow \sum_{k=0}^n \binom{n}{k} \left (\sqrt[n]{\prod_{i=1}^{n}a_i} \right )^k \left (\sqrt[n]{\prod_{i=1}^{n}b_i}\right )^{n-k} \le (a_1+b_1)(a_2+b_2)...(a_n+b_n)$$$$\Leftrightarrow \left (\sqrt[n]{\prod_{i=1}^{n}a_i}+\sqrt[n]{\prod_{i=1}^{n}b_i} \right)^n \le (a_1+b_1)(a_2+b_2)...(a_n+b_n)$$$$\Leftrightarrow \left (a_1a_2...a_n \right)^{\frac{1}{n}}+\left(b_1b_2...b_n \right)^{\frac{1}{n}}\le ((a_1+b_1)(a_2+b_2)...(a_n+b_n))^{\frac{1}{n}} $$
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