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
Number Theory
AnhQuang_67   3
N a minute ago by GreekIdiot
Source: HSGSO 2024
Let $p$ be an odd prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
3 replies
AnhQuang_67
2 hours ago
GreekIdiot
a minute ago
J
H
Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback
Blackhole.LightKing   3
N 9 minutes ago by DottedCaculator
Hi everyone,

I’ve been experimenting with a different way of approaching mathematical problem solving — a framework that emphasizes recursive structures and symbolic alignment rather than conventional step-by-step strategies.

Using this method, I recently attempted all six problems from IMO 2024 and was able to arrive at what I believe are valid full-mark solutions across the board (42/42 total score, by standard grading).

However, I don’t come from a formal competition background, so I’m sure there are gaps in clarity, communication, or even logic that I’m not fully aware of.

If anyone here is willing to take a look and provide feedback, I’d appreciate it — especially regarding:

The correctness and completeness of the proofs

Suggestions on how to make the ideas clearer or more elegant

Whether this approach has any broader potential or known parallels

I'm here to learn more and improve the presentation and thinking behind the work.

You can download the Solution here.

https://agi-origin.com/assets/pdf/AGI-Origin_IMO_2024_Solution.pdf


Thanks in advance,
— BlackholeLight0


3 replies
+1 w
Blackhole.LightKing
3 hours ago
DottedCaculator
9 minutes ago
J
H
2 var inequalities
sqing   3
N 12 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$ \frac{ a + b }{ a^2(1+ b^2)} \leq \sqrt 5-1$$$$ \frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \frac{3(\sqrt5-1)}{2}$$$$ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq2$$Solution:
$a\ge\frac{b}{2b-1}, b>\frac12$ and $ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le\frac{2ab+a^2b^2}{a^2(1+b^2)}=1+\frac{2b-a}{a(1+b^2)} \le 1+\frac{4b-3}{b^2+1}$

Assume $u=4b-3>0$ then $ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le 1+\frac{16u}{u^2+6u+25} =2+ \frac{16}{6+u+\frac{25}u} \le 3$
Equalityholds when $a=\frac{2}{3},b=2. $
3 replies
sqing
Yesterday at 1:13 PM
sqing
12 minutes ago
J
H
hard problem
Cobedangiu   8
N 12 minutes ago by ReticulatedPython
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
8 replies
Cobedangiu
Apr 21, 2025
ReticulatedPython
12 minutes ago
J
No more topics!
H
J
H
Two Functional Inequalities
Mathdreams   7
N Apr 7, 2025 by John_Mgr
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
7 replies
Mathdreams
Apr 6, 2025
John_Mgr
Apr 7, 2025
J
Two Functional Inequalities
G H J
Reveal topic tags
functional inequalities
algebra
functional equation
L
G H BBookmark kLocked kLocked NReply
Source: 2025 Nepal Mock TST Day 2 Problem 2
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Mathdreams
1466 posts
Mathdreams
#1 h  Apr 6, 2025, 1:34 PM • 4 Y
Y by PikaPika999, AlexCenteno2007, cubres, khan.academy
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
grupyorum
1413 posts
grupyorum
#2 h  Apr 6, 2025, 1:45 PM • 2 Y
Y by PikaPika999, cubres
I will show $f(x)=x^3$ for all $x$, which clearly works.

Taking $x=y=0$ in both, we find $f(0)\le 0$ and $f(0)\ge 0$, so $f(0)=0$. Taking $y=-x$ in the second, we find $0=f(0)\le f(x)+f(-x)$. Likewise, $f(x)\le x^3$ and $f(-x)\le (-x)^3=-x^3$. So, $f(x)+f(-x)\le 0$. Combining, $f(x)+f(-x)=0$ for all $x$. Lastly, using $f(-x)=-f(x)$, we get $-x^3 \ge f(-x)=-f(x)$, so $f(x)\ge x^3$ too. Thus, $f(x)=x^3$ for all $x$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
kokcio
69 posts
kokcio
#3 h  Apr 6, 2025, 1:46 PM • 2 Y
Y by PikaPika999, cubres
Putting $x=y=0$, we have $f(0)\leq 2f(0)$, but $f(0)\leq 0$, so we have to have that $f(0)=0$.
Putting now $y=-x$, we have $0\leq f(x)+f(-x) \leq x^3 + (-x)^3 = 0$, but this means that we have to have $f(x)=x^3$, because of inequalities.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
John_Mgr
67 posts
John_Mgr
#4 h  Apr 6, 2025, 4:33 PM • 2 Y
Y by PikaPika999, cubres
Claim: $f(x) \equiv x^3$ is the only function $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfies the given conditions.
Let P(x,y) be the assertion for the convenience.
i.e $f(x+y) \le f(x)+f(y)+3xy(x+y)$
P(0,0): $f(0)\ge0$ and $f(0)\le0$, $\Rightarrow$ $f(0)=0$,
Assume $f(x)=g(x)+x^3$, $\implies$ $g(x+y)\le g(x)+g(y)$
This tells us that g is subadditive.
$g(x)\le 0$ as $f(x)\le x^3$
Let Q(x,y) be the assertion for $g(x+y)\le g(x)+g(y)$
Q(x,-x): $g(0)\le g(x)+g(-x)$$\rightarrow g(x)+g(-x)\ge 0$ as $f(0)=0$ but $g(x)+g(-x)\le 0$
So, $g(x)+g(-x)=0$$\Rightarrow g(-x)=-g(x) \ge 0$
The only way both $g(x)\le x$ and $g(-x)\ge 0$ hold is if $g(x)=0$, $\forall$ $x\in \mathbb{R}$
Therefore $\boxed{f(x)=x^3}$, $\forall$ $x\in \mathbb{R}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jasperE3
11241 posts
jasperE3
#5 h  Apr 6, 2025, 7:01 PM • 5 Y
Y by PikaPika999, Assassino9931, megarnie, navier3072, cubres
Let $g(x)=f(x)-x^3+x$, it reduces to this old problem:
https://artofproblemsolving.com/community/c6h257315p1403063
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Maximilian113
554 posts
Maximilian113
#6 h  Apr 6, 2025, 7:16 PM • 2 Y
Y by PikaPika999, cubres
Note that $x=y=0 \implies 0 \leq f(0) \leq 0 \implies f(0)=0.$ Now $$y=-x \implies 0 \leq f(x)+f(-x) \leq f(x) -x^3 \implies x^3 \leq f(x) \implies f(x)=x^3$$for all $x,$ which clearly works.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Assassino9931
1251 posts
Assassino9931
#7 h  Apr 6, 2025, 10:18 PM • 1 Y
Y by cubres
jasperE3 wrote:
Let $g(x)=f(x)-x^3+x$, it reduces to this old problem:
https://artofproblemsolving.com/community/c6h257315p1403063

Nice catch haha
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
John_Mgr
67 posts
John_Mgr
#8 h  Apr 7, 2025, 3:59 AM • 1 Y
Y by cubres
jasperE3 wrote:
Let $g(x)=f(x)-x^3+x$, it reduces to this old problem:
https://artofproblemsolving.com/community/c6h257315p1403063

Good one!!
Z K Y
N Quick Reply
G
H
=
a