Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

We have your learning goals covered with Spring and Summer courses available. Enroll today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
a My Retirement & New Leadership at AoPS
rrusczyk   1271
N 4 minutes ago by tara18
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1271 replies
+5 w
rrusczyk
Monday at 6:37 PM
tara18
4 minutes ago
J
H
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Sunday, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
Sunday, Apr 13 - Aug 10
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

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


Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Mar 23 - Jul 20
Monday, Apr 7 - Jul 28
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Mar 16 - Jun 8
Wednesday, Apr 16 - Jul 2
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

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

Introduction to Algebra B Self-Paced

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

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

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
Monday, Apr 21 - Oct 13
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Sunday, Mar 23 - Aug 3
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

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

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

Advanced: Grades 9-12

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

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

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

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

MATHCOUNTS/AMC 8 Advanced
Friday, Apr 11 - Jun 27
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

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

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

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Monday, Mar 24 - Jun 16
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Mar 2, 2025
0 replies
J
H
k i Suggestion Form
jwelsh   0
May 6, 2021
Hello!

Given the number of suggestions we’ve been receiving, we’re transitioning to a suggestion form. If you have a suggestion for the AoPS website, please submit the Google Form:
Suggestion Form

To keep all new suggestions together, any new suggestion threads posted will be deleted.

Please remember that if you find a bug outside of FTW! (after refreshing to make sure it’s not a glitch), make sure you’re following the How to write a bug report instructions and using the proper format to report the bug.

Please check the FTW! thread for bugs and post any new ones in the For the Win! and Other Games Support Forum.
0 replies
jwelsh
May 6, 2021
0 replies
J
H
k i Read me first / How to write a bug report
slester   3
N May 4, 2019 by LauraZed
Greetings, AoPS users!

If you're reading this post, that means you've come across some kind of bug, error, or misbehavior, which nobody likes! To help us developers solve the problem as quickly as possible, we need enough information to understand what happened. Following these guidelines will help us squash those bugs more effectively.

Before submitting a bug report, please confirm the issue exists in other browsers or other computers if you have access to them.

For a list of many common questions and issues, please see our user created FAQ, Community FAQ, or For the Win! FAQ.

What is a bug?
A bug is a misbehavior that is reproducible. If a refresh makes it go away 100% of the time, then it isn't a bug, but rather a glitch. That's when your browser has some strange file cached, or for some reason doesn't render the page like it should. Please don't report glitches, since we generally cannot fix them. A glitch that happens more than a few times, though, could be an intermittent bug.

If something is wrong in the wiki, you can change it! The AoPS Wiki is user-editable, and it may be defaced from time to time. You can revert these changes yourself, but if you notice a particular user defacing the wiki, please let an admin know.

The subject
The subject line should explain as clearly as possible what went wrong.

Bad: Forum doesn't work
Good: Switching between threads quickly shows blank page.

The report
Use this format to report bugs. Be as specific as possible. If you don't know the answer exactly, give us as much information as you know. Attaching a screenshot is helpful if you can take one.

Summary of the problem:
Page URL:
Steps to reproduce:
1.
2.
3.
...
Expected behavior:
Frequency:
Operating system(s):
Browser(s), including version:
Additional information:

If your computer or tablet is school issued, please indicate this under Additional information.

Example
Summary of the problem: When I click back and forth between two threads in the site support section, the content of the threads no longer show up. (See attached screenshot.)
Page URL: http://artofproblemsolving.com/community/c10_site_support
Steps to reproduce:
1. Go to the Site Support forum.
2. Click on any thread.
3. Click quickly on a different thread.
Expected behavior: To see the second thread.
Frequency: Every time
Operating system: Mac OS X
Browser: Chrome and Firefox
Additional information: Only happens in the Site Support forum. My tablet is school issued, but I have the problem at both school and home.

How to take a screenshot
Mac OS X: If you type ⌘+Shift+4, you'll get a "crosshairs" that lets you take a custom screenshot size. Just click and drag to select the area you want to take a picture of. If you type ⌘+Shift+4+space, you can take a screenshot of a specific window. All screenshots will show up on your desktop.

Windows: Hit the Windows logo key+PrtScn, and a screenshot of your entire screen. Alternatively, you can hit Alt+PrtScn to take a screenshot of the currently selected window. All screenshots are saved to the Pictures → Screenshots folder.

Advanced
If you're a bit more comfortable with how browsers work, you can also show us what happens in the JavaScript console.

In Chrome, type CTRL+Shift+J (Windows, Linux) or ⌘+Option+J (Mac).
In Firefox, type CTRL+Shift+K (Windows, Linux) or ⌘+Option+K (Mac).
In Internet Explorer, it's the F12 key.
In Safari, first enable the Develop menu: Preferences → Advanced, click "Show Develop menu in menu bar." Then either go to Develop → Show Error console or type Option+⌘+C.

It'll look something like this:
IMAGE
3 replies
slester
Apr 9, 2015
LauraZed
May 4, 2019
J
H
k i Community Safety
dcouchman   0
Jan 18, 2018
If you find content on the AoPS Community that makes you concerned for a user's health or safety, please alert AoPS Administrators using the report button (Z) or by emailing sheriff@aops.com . You should provide a description of the content and a link in your message. If it's an emergency, call 911 or whatever the local emergency services are in your country.

Please also use those steps to alert us if bullying behavior is being directed at you or another user. Content that is "unlawful, harmful, threatening, abusive, harassing, tortuous, defamatory, vulgar, obscene, libelous, invasive of another's privacy, hateful, or racially, ethnically or otherwise objectionable" (AoPS Terms of Service 5.d) or that otherwise bullies people is not tolerated on AoPS, and accounts that post such content may be terminated or suspended.
0 replies
dcouchman
Jan 18, 2018
0 replies
J
H
2025 Caucasus MO Juniors P7
BR1F1SZ   2
N a minute ago by doongus
Source: Caucasus MO
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$a right-angled triangle can be formed?
2 replies
BR1F1SZ
an hour ago
doongus
a minute ago
J
H
divisors on a circle
Valentin Vornicu   46
N 3 minutes ago by doongus
Source: USAMO 2005, problem 1, Zuming Feng
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
46 replies
+1 w
Valentin Vornicu
Apr 21, 2005
doongus
3 minutes ago
J
H
Maximum of Incenter-triangle
mpcnotnpc   1
N 34 minutes ago by mpcnotnpc
Triangle $\Delta ABC$ has side lengths $a$, $b$, and $c$. Select a point $P$ inside $\Delta ABC$, and construct the incenters of $\Delta PAB$, $\Delta PBC$, and $\Delta PAC$ and denote them as $I_A$, $I_B$, $I_C$. What is the maximum area of the triangle $\Delta I_A I_B I_C$?
1 reply
mpcnotnpc
Yesterday at 6:24 PM
mpcnotnpc
34 minutes ago
J
H
IMO 2009, Problem 2
orl   141
N 36 minutes ago by mananaban
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
141 replies
orl
Jul 15, 2009
mananaban
36 minutes ago
J
H
Pressing 'go down button' always creates a gray box on the last post
Craftybutterfly   20
N Yesterday at 12:03 AM by Craftybutterfly
Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post
Page URL: any forum
Steps to reproduce:
1. Go to any topic in a forum
2. The gray box at the bottom overlaps part of the first post
Expected behavior: Should not show a gray box
Frequency: 100% of the time
Operating system(s): Linux HP EliteBook 835 G8 Notebook PC
Browser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)
Additional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears
20 replies
Craftybutterfly
Mar 12, 2025
Craftybutterfly
Yesterday at 12:03 AM
J
H
k The avatars are not consistent
Craftybutterfly   45
N Monday at 11:03 PM by Demetri
Summary of the problem: The avatars are not consistent
Page URL: idk
Steps to reproduce:
1. change your avatar
2.reload a topic you posted in
3. do #2 to a different topic with your post in it
Expected behavior: Avatars are the same
Frequency: 100%
Operating system(s): MacOS
Browser(s), including version: Chrome latest version
Additional information: refreshing does not help, neither does logging out and in
45 replies
Craftybutterfly
Mar 20, 2025
Demetri
Monday at 11:03 PM
J
H
k Mr. Rusczyk is retiring!
SmartGroot   58
N Monday at 8:33 PM by AmethystC
Has anyone else got the email? Mr. Rusczyks retiring :o
58 replies
SmartGroot
Monday at 4:12 PM
AmethystC
Monday at 8:33 PM
J
H
Python exit() module decriptions appear as "undefined"
SoaringHigh   8
N Mar 22, 2025 by Major_Monogram
Summary of the problem: When using exit() (or quit()) in the Python windows on AoPS the "Description" and "To fix" options show up as "undefined"
sample program
Page URL: N/A
Steps to reproduce:
1. Use the AoPS Python module to execute the exit() or quit() functions in a program. (try running the sample program)
Expected behavior: The "Description" and "To fix" sections give a description of SystemExit
Frequency: Always
Operating system(s): Windows 11 Home
Browser(s), including version: Microsoft Edge 130.0.2849.46
Additional information: N/A
8 replies
SoaringHigh
Oct 22, 2024
Major_Monogram
Mar 22, 2025
J
H
Help about topics on alcumus
GreenBanana666   10
N Mar 22, 2025 by Major_Monogram
My ratio basics score used to be 99 a 5 days ago. But it is 84 now. What and why did that happen?


Now after one hour and zero questions done on Ratio basics, it has risen by 5 rating. :ewpu:
10 replies
GreenBanana666
Oct 29, 2024
Major_Monogram
Mar 22, 2025
J
H
k Happy 1795^2 day
maxamc   2
N Mar 22, 2025 by cheltstudent
$30222025=1795^2$.

$3^2$ states are taking mathcounts state and $1^2$ territories are taking mathcounts state ($6^2$, $2^2$ national qualifiers each).

$5^2$ perfect square dates this century, $5$ this year.
2 replies
maxamc
Mar 22, 2025
cheltstudent
Mar 22, 2025
J
H
k AMC 12a 2024 Problems and Solutions Pages Misaligned?
ViolinGod   5
N Mar 21, 2025 by jlacosta
I was looking at the past AMC problems and it seems like the problems listed on this page:

https://artofproblemsolving.com/wiki/index.php/2024_AMC_12A_Problems

have solution links that do not all correspond to the correct problem. For example, for problem 2, the solution page is linked as:

https://artofproblemsolving.com/wiki/index.php/2024_AMC_12A_Problems/Problem_2, but in the first link the problems are different.
5 replies
ViolinGod
Mar 21, 2025
jlacosta
Mar 21, 2025
J
H
k Number of posts is not consistent
LawofCosine   3
N Mar 21, 2025 by LawofCosine
The number of posts is not consistent. see attachments. Thank you.
3 replies
LawofCosine
Mar 21, 2025
LawofCosine
Mar 21, 2025
J
H
k How to add members to forums?
Bnn81351   2
N Mar 20, 2025 by Craftybutterfly
How do I add members to forums?
2 replies
Bnn81351
Mar 20, 2025
Craftybutterfly
Mar 20, 2025
J
H
k AoPS Academy: Exporting rich format results in broken BBCode.
Minium   1
N Mar 19, 2025 by jlacosta
When exporting a rich format document in the writing tab into the message board, bold formatting specifically is broken and results in broken BBCode.
Page URL: virtual.aopsacademy.org/class/<any writing class works here>/writing

TO REPRODUCE
1. enter "Lorem ipsum".
2. apply bold to "Lorem"
3. apply italic to "ip".
4. click the Post Draft on Message Board
5. read the contents of the message board post.

FOR EXAMPLE
When I format "Lorem ipsum" (in the writing tab of course), but when I export to post it, I get

[code]Lorem[/b] ipsum[/code].

Notice that the first bolding does not start, only ends.
1 reply
Minium
Mar 18, 2025
jlacosta
Mar 19, 2025
J
H
J
H
Find all functions
Jackson0423   2
N Monday at 5:19 AM by pco
Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..
2 replies
Jackson0423
Mar 23, 2025
pco
Monday at 5:19 AM
J
Find all functions
G H J
Reveal topic tags
function
algebra
L
G H BBookmark kLocked kLocked NReply
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Jackson0423
1 post
Jackson0423
#1 h  Mar 23, 2025, 4:06 PM
Y by
Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
pco
23474 posts
pco
#2 h  Monday at 5:08 AM
Y by
Jackson0423 wrote:
Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..
No : $x+\frac 1x$ is undefined when $x=0$ and so is not a function from $\mathbb R\to\mathbb R$ satisfying functional equation $\frac 1{f(f(x))-f(x)}=f(x)$ $\forall x\in\mathbb R$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
pco
23474 posts
pco
#3 h  Monday at 5:19 AM
Y by
Jackson0423 wrote:
Find all functions F:R->R such that
1/(F(F(x))-F(x))=F(x)
I know x+1/x works..
In fact ou have $f(x)=x+\frac 1x$ $\forall x\in A=f(\mathbb R)$

And so general solution :
Let any $A\subseteq\mathbb R\setminus \{0\}$ stable by operation $x+\frac 1x$ ($a+\frac 1a\in A$ $\forall a\in A$)
Let $g(x)$ any function from $\mathbb R\to A$

$\forall x\in A$ : $f(x)=x+\frac 1x$
$\forall x\notin A$ : $f(x)=g(x)$

Example of such $A$ : $\mathbb R\setminus\{0\}$, $\mathbb Q\setminus\{0\}$, the set of all algebraic nonzero numbers, and a lot of others ....
Z K Y
N Quick Reply
G
H
=
a