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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.

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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

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

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

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

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

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

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

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

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

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

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

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

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

Advertising paid program or service: never allowed

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

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

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

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
Nesbitt in triangle
Dynic   0
39 minutes ago
Let $a,b,c$ be the lengths of the three sides of a triangle, respectively. Let $\mathbb{X}$ as a set of positive real number and the function $D:\mathbb{X}^3\to\mathbb{X}$ determined by $$D(x,y,z)=\dfrac{x}{y+z}+\dfrac{y}{x+z}+\dfrac{z}{x+y},\forall x,y,z >0$$Find all values that $D(a,b,c)$ can take.
0 replies
Dynic
39 minutes ago
0 replies
Geometry / Ellipses
SpacePhysics   0
42 minutes ago
From three points A,B,C fixed in space is formed an acute triangle ABC, and three ellipses such that the points A,B,C are their foci, and they all intersect in a point P inside the triangle. Find P such that the sum of the semi-major axis from the three ellipses is minimized
0 replies
SpacePhysics
42 minutes ago
0 replies
a+b+c=3 ine
jokehim   0
42 minutes ago
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
0 replies
1 viewing
jokehim
42 minutes ago
0 replies
Inequalities
sqing   3
N an hour ago by sqing
Let $ a,b\geq 2  . $ Prove that
$$ (1-a^2)(1-b^2) (ab-1)^2-108  ab \geq-351 $$$$(1-a^2)(1-b^2) (ab-1)(a^2b^2-1)-  621 ab \geq-2079 $$
3 replies
sqing
an hour ago
sqing
an hour ago
[Registration Open] Mustang Math Tournament 2025
MustangMathTournament   31
N Today at 4:00 AM by ethan2011
Mustang Math is excited to announce that registration for our annual tournament, MMT 2025, is open! This year, we are bringing our tournament to 9 in-person locations, as well as online!

Locations include: Colorado, Norcal, Socal, Georgia, Illinois, Massachusetts, New Jersey, Nevada, Washington, and online. For registration and more information, check out https://mustangmath.com/competitions/mmt-2025.

MMT 2025 is a math tournament run by a group of 150+ mathematically experienced high school and college students who are dedicated to providing a high-quality and enjoyable contest for middle school students. Our tournament centers around teamwork and collaboration, incentivizing students to work with their teams not only to navigate the challenging and interesting problems of the tournament but also to develop strategies to master the unique rounds. This includes a logic puzzle round, a strategy-filled hexes round, a race-like gallop round, and our trademark ‘Mystery Mare’ round!

Awards:
[list]
[*] Medals for the top teams
[*] Shirts, pins, stickers and certificates for all participants
[*] Additional awards provided by our wonderful sponsors!
[/list]

We are also holding a free MMT prep seminar from 3/15-3/16 to help students prepare for the upcoming tournament. Join the Google Classroom! https://classroom.google.com/c/NzQ5NDUyNDY2NjM1?cjc=7sogth4
31 replies
MustangMathTournament
Mar 8, 2025
ethan2011
Today at 4:00 AM
JSMCR Results
FuturePanda   14
N Today at 2:40 AM by Pengu14
Hi everyone,

Did anyone get their JSMCR decisions back yet? They were supposed to release on 2/28

Thanks!
14 replies
FuturePanda
Mar 1, 2025
Pengu14
Today at 2:40 AM
AMC 10.........
BAM10   8
N Today at 2:19 AM by ChickensEatGrass
I'm in 8th grade and have never taken the AMC 10. I am currently in alg2. I have scored 20 on AMC 8 this year and 34 on the chapter math counts last year. Can I qualify for AIME. Also what should I practice AMC 10 next year?
8 replies
BAM10
Mar 2, 2025
ChickensEatGrass
Today at 2:19 AM
Tennessee Math Tournament (TMT) Online 2025
TennesseeMathTournament   31
N Today at 2:16 AM by NashvilleSC
Hello everyone! We are excited to announce a new competition, the Tennessee Math Tournament, created by the Tennessee Math Coalition! Anyone can participate in the virtual competition for free.

The testing window is from March 22nd to April 5th, 2025. Virtual competitors may participate in the competition at any time during that window.

The virtual competition consists of three rounds: Individual, Bullet, and Team. The Individual Round is 60 minutes long and consists of 30 questions (AMC 10 level). The Bullet Round is 20 minutes long and consists of 80 questions (Mathcounts Chapter level). The Team Round is 30 minutes long and consists of 16 questions (AMC 12 level). Virtual competitors may compete in teams of four, or choose to not participate in the team round.

To register and see more information, click here!

If you have any questions, please email connect@tnmathcoalition.org or reply to this thread!
31 replies
TennesseeMathTournament
Mar 9, 2025
NashvilleSC
Today at 2:16 AM
hcssim application question
enya_yurself   2
N Today at 1:35 AM by Magnetoninja
do they send the Interesting Test to everyone who applied or do they read the friendly letter first and only send to the kids they like?
2 replies
enya_yurself
Yesterday at 11:13 PM
Magnetoninja
Today at 1:35 AM
The return of American geo
brianzjk   77
N Today at 1:17 AM by Ilikeminecraft
Source: USAJMO 2023/6
Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$.

Proposed by Anton Trygub
77 replies
brianzjk
Mar 23, 2023
Ilikeminecraft
Today at 1:17 AM
Apply for Team USA at the International Math Competition (IMC)!
peace09   53
N Today at 1:17 AM by stjwyl
The International Math Competition (IMC) is essentially the elementary and middle school equivalent of the IMO, with individual and team rounds featuring both short-answer and proof-based problems. See past problems here.

Team USA is looking for 6th graders and below with AIME qualification or AMC 8 DHR (or equivalent), and for 9th graders and below with JMO or Mathcounts Nationals qualification. If you think you meet said criteria, fill out the initial form here.

Here are a couple quick links for further information:
[list=disc]
[*] Dr. Tao Hong's website, which contains a detailed recap of the 2024 competition (and previous years'), as well as Team USA's historical results. (You may recognize a couple names... @channing421 @vrondoS et al.: back me up here :P)
[*] My journal, which gives an insider's perspective on the camp :ninja:
[/list]
53 replies
peace09
Aug 13, 2024
stjwyl
Today at 1:17 AM
OTIS Mock AIME 2025 airs Dec 19th
v_Enhance   39
N Today at 1:06 AM by MonkeyLuffy
Source: https://web.evanchen.cc/mockaime.html
Satisfactory. Keep cooking.
IMAGE

Problems are posted at https://web.evanchen.cc/mockaime.html#current now!

Like last year, we're running the OTIS Mock AIME 2025 again, except this time there will actually be both a I and a II because we had enough problems to pull it off. However, the two versions will feel quite different from each other:

[list]
[*] The OTIS Mock AIME I is going to be tough. It will definitely be harder than the actual AIME, by perhaps 2 to 4 problems. But more tangibly, it will also have significant artistic license. Problems will freely assume IMO-style background throughout the test, and intentionally stretch the boundary of what constitutes an “AIME problem”.
[*] The OTIS Mock AIME II is meant to be more practically useful. It will adhere more closely to the difficulty and style of the real AIME. There will inevitably still be some more IMO-flavored problems, but they’ll appear later in the ordering.
[/list]
Like last time, all 30 problems are set by current and past OTIS students.

Details are written out at https://web.evanchen.cc/mockaime.html, but to highlight important info:
[list]
[*] Free, obviously. Anyone can participate.
[*]Both tests will release sometime Dec 19th. You can do either/both.
[*]If you'd like to submit for scoring, you should do so by January 20th at 23:59 Pacific time (same deadline for both). Please hold off on public spoilers before then.
[*]Solutions, statistics, and maybe some high scores will be published shortly after that.
[/list]
Feel free to post questions, hype comments, etc. in this thread.
39 replies
1 viewing
v_Enhance
Dec 6, 2024
MonkeyLuffy
Today at 1:06 AM
They mixed up USAJMO and AIME I guess
Math4Life7   54
N Today at 12:36 AM by littlefox_amc
Source: USAJMO 2024/1
Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral.

Proposed by Evan O'Dorney
54 replies
Math4Life7
Mar 20, 2024
littlefox_amc
Today at 12:36 AM
USAMO vs USAJMO Prestige
elasticwealth   47
N Today at 12:36 AM by axusus
Just curious, what does everyone think about the prestige of a USAMO qualification vs a USAJMO qual? Obv USAMO > USAJMO but how much?

And while we’re at it, what about amc 10 dhr vs amc 12 dhr? Thoughts?

Edit: also how much is a perfect score on 10/12 worth? Asking for a friend….
47 replies
elasticwealth
Feb 17, 2025
axusus
Today at 12:36 AM
(x^2-3x+2)^2-3(x^2-3x+2)-2-x=0 (Moldova 2000 Grade 9 P5)
jasperE3   12
N Apr 26, 2021 by R-sk
Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
12 replies
jasperE3
Apr 26, 2021
R-sk
Apr 26, 2021
(x^2-3x+2)^2-3(x^2-3x+2)-2-x=0 (Moldova 2000 Grade 9 P5)
G H J
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jasperE3
11096 posts
#1
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Solve in real numbers the equation
$$\left(x^2-3x-2\right)^2-3\left(x^2-3x-2\right)-2-x=0.$$
This post has been edited 1 time. Last edited by jasperE3, Apr 26, 2021, 1:01 PM
Reason: x^2-3x-2 instead of x^2-3x+2
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Mathematician1010
3261 posts
#2
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Start out by simplifying the left-hand side, and you get
$x^4-6x^3+10x^2-4x-4=0$
After that, I'd recommend using the rational root theorem, and if there are any rational solutions, you can divide out the factor. I haven't done this for this problem yet, but if there are two rational solutions you can reduce down to a quadratic and find any other solutions with the quadratic formula.
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jasperE3
11096 posts
#3
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RRT gives no rational roots, you'd probably have to factor it into two quadratics. Any solutions without undetermined coefficients?
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Lamboreghini
6486 posts
#4
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progress
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jasperE3
11096 posts
#5 • 3 Y
Y by Mango247, Mango247, Mango247
yofro (via PM) wrote:
What happens if x is a root of $x^2-4x-2$?
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Lamboreghini
6486 posts
#6
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Post #5 by jasperE3

@jasperE3 ok

soo... some more progress
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natmath
8219 posts
#7
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jasperE3 wrote:
yofro (via PM) wrote:
What happens if x is a root of $x^2-4x-2$?

That's pretty cool
Click to reveal hidden text

The $x^2-4x-2$ is kind of intuitive because clearly setting $x^2-3x-2=x$ gives a solution to the above equation. We still need the other quadratic factor for 4 roots, but I'm not seeing something that is intuitive. Of course, it's pretty easy now to just factor $x^2-4x-2$ from the equation in #2 and use MUD, but I was wondering if there was a more intuitive approach on that other factor.

@below you really should be thanking yofro for these amazing hints. Do you have a reason to believe the other factor does not have real roots?
This post has been edited 2 times. Last edited by natmath, Apr 26, 2021, 4:55 AM
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Lamboreghini
6486 posts
#8 • 3 Y
Y by Mango247, Mango247, Mango247
Post #7 by natmath

@natmath whoa that's awesome

solutions to $x^2-4x-2$ are $$x=\frac{4\pm\sqrt{16+8}}{2}=\frac{4\pm2\sqrt6}{2}=2\pm\sqrt6.$$Those are real! 2 solutions to the equation already!

If I'm not mistaken, these are the only real solutions and this problem is solved....?
This post has been edited 1 time. Last edited by Lamboreghini, Apr 26, 2021, 4:43 AM
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asbodke
1914 posts
#9
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uh I think you messed up? You're confusing $x^2-4x+2$ and $x^2-4x-2$? Is there a typo in the problem?

If there's a typo in the problem and it is $x^2-3x-2$ instead of $x^2-3x+2$ we can use poly division and get our other answers to be $1\pm \sqrt 5$

and of course using the other roots, we can let $x$ be a root of $x^2-2x-4$, then $x^2-3x-2=-x+2$, and $(-x+2)^2-3(-x+2)-2-x=x^2-4x+4+3x-6-2-x=x^2-2x-4=0$
This post has been edited 4 times. Last edited by asbodke, Apr 26, 2021, 5:33 AM
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Mathematician1010
3261 posts
#10
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@above where are you getting $x^2-2x-4$ from?
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jasperE3
11096 posts
#11
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Very sorry. Yes, the problem was incorrect. It has been edited now.
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natmath
8219 posts
#12 • 1 Y
Y by yofro
Yofro gave me a hint on $x^2-2x-4$. It's not as direct as the first one, but I can try to explain it.

If $f(x)=x^2-3x-2$, then we want to find the solutions to
$$f(f(x))=x$$
Of course, if there existed an $r$ s.t. $f(r)=r$, then this would clearly be a solution to our equation. That $r$ must be either of the roots to
$$x^2-3x-2=x$$$$x^2-4x-2=0$$
Let's say for some constant $k$ there exists an $r$ s.t. $f(r)=k-r$. Now let's say $k-r$ also satisfies the equation $f(x)=k-x$ (i.e. $f(k-r)=k-(k-r)=r$). Then this $r$ would satisfy the original equation.
However, finding this root is not as immediate. We want $r$ and $k-r$ to be the solutions of
$$f(x)=k-x$$$$x^2-3x-2=k-x$$$$x^2-2x-(2+k)=0$$By viete's, the sum of roots is
$$r+k-r=2$$$$k=2$$So the solutions to
$$x^2-2x-4=0$$also satisfy our original equation.

Also I think we know that there are no other solutions aside from those in the form $f(x)=x$ and $f(x)=k-x$ because $x,k-x$ are the only polynomial functions that satisfy $f(f(x))=x$.
This post has been edited 1 time. Last edited by natmath, Apr 26, 2021, 3:29 PM
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R-sk
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#13
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Take one x in other side and set y=$x^2-3x$ and solve quadratic take one of its root equate it to x then you get a good answer
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