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

Stay ahead of learning milestones! Enroll in a class over the summer!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
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.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
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
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
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
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
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
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
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
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
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
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
Tuesday, Jun 10 - Aug 26

Calculus
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
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
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
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
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
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
Apr 2, 2025
0 replies
J
H
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
H
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
J
H
nice geo
Melid   1
N 6 minutes ago by Melid
Source: 2025 Japan Junior MO preliminary P9
Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.
1 reply
Melid
11 minutes ago
Melid
6 minutes ago
J
H
Nice inequality
sqing   1
N 8 minutes ago by Oksutok
Source: WYX
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
1 reply
sqing
Apr 24, 2019
Oksutok
8 minutes ago
J
H
product of all integers of form i^3+1 is a perfect square
AlastorMoody   3
N 11 minutes ago by Assassino9931
Source: Balkan MO ShortList 2009 N3
Determine all integers $1 \le m, 1 \le n \le 2009$, for which
\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}
3 replies
AlastorMoody
Apr 6, 2020
Assassino9931
11 minutes ago
J
H
IMO ShortList 1998, combinatorics theory problem 1
orl   44
N 19 minutes ago by YaoAOPS
Source: IMO ShortList 1998, combinatorics theory problem 1
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
44 replies
orl
Oct 22, 2004
YaoAOPS
19 minutes ago
J
H
A game optimization on a graph
Assassino9931   2
N 20 minutes ago by dgrozev
Source: Bulgaria National Olympiad 2025, Day 2, Problem 6
Let \( X_0, X_1, \dots, X_{n-1} \) be \( n \geq 2 \) given points in the plane, and let \( r > 0 \) be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points \( X_0, X_1, \dots, X_{n-1} \), i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex \( X_i \) a non-negative real number \( r_i \), for \( i = 0, 1, \dots, n-1 \), such that $\sum_{i=0}^{n-1} r_i = 1$. Bob then selects a sequence of distinct vertices \( X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k} \) such that \( X_{i_j} \) and \( X_{i_{j+1}} \) are connected by an edge for every \( j = 0, 1, \dots, k-1 \). (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$.) Bob wins if
\[ \frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r; \]otherwise, Alice wins. Depending on \( n \), determine the largest possible value of \( r \) for which Bobby has a winning strategy.
2 replies
Assassino9931
Apr 8, 2025
dgrozev
20 minutes ago
J
H
Composite sum
rohitsingh0812   39
N 28 minutes ago by YaoAOPS
Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
39 replies
rohitsingh0812
Jun 3, 2006
YaoAOPS
28 minutes ago
J
H
Problem 1
SpectralS   145
N 30 minutes ago by IndexLibrorumProhibitorum
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Evangelos Psychas, Greece
145 replies
SpectralS
Jul 10, 2012
IndexLibrorumProhibitorum
30 minutes ago
J
H
Help me please
sealight2107   0
32 minutes ago
Let $m,n,p,q$ be positive reals such that $m+n+p+q+\frac{1}{mnpq} = 18$. Find the minimum and maximum value of $m,n,p,q$
0 replies
1 viewing
sealight2107
32 minutes ago
0 replies
J
H
Rectangular line segments in russia
egxa   2
N 35 minutes ago by mohsen
Source: All Russian 2025 9.1
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
2 replies
egxa
Apr 18, 2025
mohsen
35 minutes ago
J
H
(help urgent) Classic Geo Problem / Angle Chasing?
orangesyrup   4
N 35 minutes ago by Royal_mhyasd
Source: own
In the given figure, ABC is an isosceles triangle with AB = AC and ∠BAC = 78°. Point D is chosen inside the triangle such that AD=DC. Find the measure of angle X (∠BDC).

ps: see the attachment for figure
4 replies
orangesyrup
an hour ago
Royal_mhyasd
35 minutes ago
J
H
Interesting combinatoric problem on rectangles
jaydenkaka   0
an hour ago
Source: Own
Define act <Castle> as following:
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.

Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)

Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
0 replies
jaydenkaka
an hour ago
0 replies
J
H
Collect ...
luutrongphuc   2
N an hour ago by megarnie
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that:
$$f\left(f(xy)+1\right)=xf\left(x+f(y)\right)$$
2 replies
luutrongphuc
Apr 21, 2025
megarnie
an hour ago
J
H
hard problem
Cobedangiu   8
N an hour ago by IceyCold
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
8 replies
Cobedangiu
Apr 2, 2025
IceyCold
an hour ago
J
H
Factor of P(x)
Brut3Forc3   20
N an hour ago by IceyCold
Source: 1976 USAMO Problem 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\] prove that $ x-1$ is a factor of $ P(x)$.
20 replies
Brut3Forc3
Apr 4, 2010
IceyCold
an hour ago
J
H
J
H
Functional equations
hanzo.ei   20
N Apr 17, 2025 by hanzo.ei
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
20 replies
hanzo.ei
Mar 29, 2025
hanzo.ei
Apr 17, 2025
J
Functional equations
G H J
Reveal topic tags
algebra
functional equation
L
G H BBookmark kLocked kLocked NReply
Source: Greekldiot
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanzo.ei
20 posts
hanzo.ei
#1 h  Mar 29, 2025, 4:33 PM
Y by
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#2 h  Mar 29, 2025, 5:16 PM
Y by
Not my problem :D Havent made such a beautiful FE myself yet.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Mathzeus1024
826 posts
Mathzeus1024
#3 h  Mar 30, 2025, 10:12 AM
Y by
It works for $\textcolor{red}{f(x)=\frac{1}{x}}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#4 h  Mar 30, 2025, 10:23 AM
Y by
Yeah we established that in another post
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanzo.ei
20 posts
hanzo.ei
#5 h  Mar 30, 2025, 10:42 AM
Y by
pco, can you solve it :omighty:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#6 h  Mar 30, 2025, 11:27 AM
Y by
$f$ seems to be an involution. I wonder if we are able to prove that. Then we can eliminate other solutions to the assertion very easily.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanzo.ei
20 posts
hanzo.ei
#7 h  Apr 2, 2025, 3:28 PM
Y by
bump!!!!
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MR.1
102 posts
MR.1
#8 h  Apr 3, 2025, 4:08 PM • 3 Y
Y by hanzo.ei, Akakri, giangtruong13
solved with GioOrnikapa if you guys want solution please give me $10$ likes :-D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#9 h  Apr 3, 2025, 4:17 PM
Y by
lol aint no way
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GioOrnikapa
76 posts
GioOrnikapa
#10 h  Apr 3, 2025, 4:33 PM • 1 Y
Y by MR.1
A lot of liars nowadays smh
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
truongphatt2668
323 posts
truongphatt2668
#11 h  Apr 3, 2025, 4:34 PM
Y by
hanzo.ei wrote:
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

I can prove $f(x)$ is injective and $f(1)=1$ anyone continue please?
This post has been edited 3 times. Last edited by truongphatt2668, Apr 3, 2025, 4:52 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#12 h  Apr 3, 2025, 5:55 PM
Y by
truongphatt2668 wrote:
hanzo.ei wrote:
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

I can prove $f(x)$ is injective and $f(1)=1$ anyone continue please?

I noticed that there exists some homogenous-like function by isolating $y$ on the $RHS$. Can you post the claims you made with proof so that we can create a complete solution?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
truongphatt2668
323 posts
truongphatt2668
#13 h  Apr 4, 2025, 1:42 AM
Y by
GreekIdiot wrote:
truongphatt2668 wrote:
hanzo.ei wrote:
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

I can prove $f(x)$ is injective and $f(1)=1$ anyone continue please?

I noticed that there exists some homogenous-like function by isolating $y$ on the $RHS$. Can you post the claims you made with proof so that we can create a complete solution?

Just do it, and I will give a complete solution :D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#14 h  Apr 4, 2025, 3:29 PM
Y by
$f(x)>0$ for all $x \in \mathbb R_+$ so all $y \in \mathbb R_+$ can be written as $\dfrac {f(m)}{f(n)}$ for some $m,n \in \mathbb R_+$
Then there exists some homogenous-kinda function (lets call it $g$) such that $f(xf(y)+f(x))=y^{\ell +1} \cdot g(x)$ and also $f(x+yf(x))=y^{\ell} \cdot g(x)$ thats what I meant to say. Correct me if wrong lol. :oops:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jasperE3
11236 posts
jasperE3
#15 h  Apr 4, 2025, 4:36 PM
Y by
GreekIdiot wrote:
$f(x)>0$ for all $x \in \mathbb R_+$ so all $y \in \mathbb R_+$ can be written as $\dfrac {f(m)}{f(n)}$ for some $m,n \in \mathbb R_+$
Then there exists some homogenous-kinda function (lets call it $g$) such that $f(xf(y)+f(x))=y^{\ell +1} \cdot g(x)$ and also $f(x+yf(x))=y^{\ell} \cdot g(x)$ thats what I meant to say. Correct me if wrong lol. :oops:

What's a homogenous kinda function
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#16 h  Apr 4, 2025, 8:05 PM
Y by
I am not sure how to call it in english or even what it is. Hope you can understand what I am saying from the symbols :D Thats the important part anyways, not some random math definition.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jasperE3
11236 posts
jasperE3
#17 h  Apr 4, 2025, 8:07 PM
Y by
GreekIdiot wrote:
I am not sure how to call it in english or even what it is. Hope you can understand what I am saying from the symbols :D Thats the important part anyways, not some random math definition.

I can't, can you explain?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#18 h  Apr 4, 2025, 8:12 PM
Y by
So basically I am trying to define a second function, g, which exists and satisfies both relations above. Then proving g must be constant will help in proving that the only sol we have found so far is unique. Hope that clears things up.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jasperE3
11236 posts
jasperE3
#19 h  Apr 4, 2025, 8:15 PM
Y by
GreekIdiot wrote:
So basically I am trying to define a second function, g, which exists and satisfies both relations above. Then proving g must be constant will help in proving that the only sol we have found so far is unique. Hope that clears things up.

How is $g$ defined
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreekIdiot
174 posts
GreekIdiot
#20 h  Apr 4, 2025, 8:18 PM
Y by
$g(x) > 0$ is a must for all positive $x$. Then it could be any function but we may be able to narrow it down. Just brainstorming, nothing rigorous. This FE has been unsolved for some time, I doubt that I of all people will be the one to solve.
This post has been edited 3 times. Last edited by GreekIdiot, Apr 4, 2025, 8:28 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanzo.ei
20 posts
hanzo.ei
#21 h  Apr 17, 2025, 2:09 PM
Y by
:blush: :blush:
Z K Y
N Quick Reply
G
H
=
a