Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
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
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
Inequalities
sqing   5
N 3 minutes ago by sqing
Let $ a,b>0, a^2+ab+b^2 \geq 6  $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10}  $. Prove that
$$a^4+ab+b^4  \leq 10$$Let $ a,b>0,  a^2+ab+b^2 \geq \frac{15}{2}  $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0,  a^2+ab+b^2 \leq \sqrt{10}  $. Prove that
$$-\frac{1}{8}\leq  a^4-ab+b^4\leq 10$$
5 replies
1 viewing
sqing
Yesterday at 2:42 PM
sqing
3 minutes ago
harmonic quadrilateral
Lukariman   2
N 10 minutes ago by Lukariman
Given quadrilateral ABCD inscribed in a circle with center O. CA:CB= DA:DB are satisfied. M is any point and d is a line parallel to MC. Radial projection M transforms A,B,D onto line d into A',B',D'. Prove that B' is the midpoint of A'D'.
2 replies
Lukariman
Today at 6:36 AM
Lukariman
10 minutes ago
Find the angle alpha [Iran Second Round 1994]
Amir Hossein   4
N 11 minutes ago by Mysteriouxxx
In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$
IMAGE
4 replies
Amir Hossein
Nov 26, 2010
Mysteriouxxx
11 minutes ago
Inequality
lgx57   1
N 17 minutes ago by DAVROS
Source: Own
$a,b>0$,$a^4+a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.

$a,b>0$,$a^4-a^2b^2+b^4=k$.Find the min of $4a^2-ab+4b^2$.
1 reply
lgx57
5 hours ago
DAVROS
17 minutes ago
Calculating sum of the numbers
Sadigly   2
N 35 minutes ago by Gggvds1
Source: Azerbaijan Junior MO 2025 P4
A $3\times3$ square is filled with numbers $1;2;3...;9$.The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?

$\text{a)}$ $24,24,25,25$

$\text{b)}$ $20,23,26,29$
2 replies
+1 w
Sadigly
6 hours ago
Gggvds1
35 minutes ago
Cube Sphere
vanstraelen   0
36 minutes ago

Given the cube $\left(\begin{array}{ll} EFGH \\ ABCD \end{array}\right)$ with edge $6$ cm.
Find the volume of the sphere passing through $A,B,C,D$ and tangent to the plane $(EFGH)$.
0 replies
vanstraelen
36 minutes ago
0 replies
Help me identify what should i focus in alcumus for contest's each problem
Hope_and_fight   0
39 minutes ago

So this file i have attached is a sample test from my upcoming regional schools' math contest. This is just a test sample, i was looking for problems as similar as to these so i have more material to practice with. Unfortunately i don't have much time to read the whole books the contest is already soon. I want to test myself as much as possible with problems, also Alcumus shows from what section of the book it is. so i can kinda cram the pages

TLDR: I WOULD BE REALLY GRATEFUL IF YOU COULD POINT ON WHAT TO FOCUS ON ALCUMUS FOR EACH OF THESE CONTEST PROBLEMS. AND SORRY FOR MY ENGLISH. I DON'T KNOW MUCH OF MATH IN THERE:)
0 replies
Hope_and_fight
39 minutes ago
0 replies
How many pairs
Ecrin_eren   6
N an hour ago by Ecrin_eren


Let n be a natural number and p be a prime number. How many different pairs (n, p) satisfy the equation:

p + 2^p + 3 = n^2 ?



6 replies
Ecrin_eren
May 2, 2025
Ecrin_eren
an hour ago
Divisibility..
Sadigly   1
N an hour ago by Mathzeus1024
Source: Azerbaijan Junior MO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
1 reply
Sadigly
Today at 7:37 AM
Mathzeus1024
an hour ago
official solution of IGO
ABCD1728   7
N an hour ago by ABCD1728
Source: IGO official website
Where can I get the official solution of IGO for 2023 and 2024, there are some inhttps://imogeometry.blogspot.com/p/iranian-geometry-olympiad.html, but where can I find them on the official website, thanks :)
7 replies
ABCD1728
May 4, 2025
ABCD1728
an hour ago
Combo geo with circles
a_507_bc   10
N an hour ago by EthanWYX2009
Source: 239 MO 2024 S8
There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.
10 replies
+1 w
a_507_bc
May 22, 2024
EthanWYX2009
an hour ago
Square number
linkxink0603   2
N an hour ago by linkxink0603
Find m is positive interger such that m^4+3^m is square number
2 replies
linkxink0603
2 hours ago
linkxink0603
an hour ago
Vietnam TST #5
IMOStarter   2
N an hour ago by cursed_tangent1434
Source: Vietnam TST 2022 P5
A fractional number $x$ is called pretty if it has finite expression in base$-b$ numeral system, $b$ is a positive integer in $[2;2022]$. Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$
2 replies
IMOStarter
Apr 27, 2022
cursed_tangent1434
an hour ago
Squares consisting of digits 0, 4, 9
VicKmath7   4
N 2 hours ago by NicoN9
Source: Bulgaria MO Regional round 2024, 9.3
A positive integer $n$ is called a $\textit{supersquare}$ if there exists a positive integer $m$, such that $10 \nmid m$ and the decimal representation of $n=m^2$ consists only of digits among $\{0, 4, 9\}$. Are there infinitely many $\textit{supersquares}$?
4 replies
VicKmath7
Feb 13, 2024
NicoN9
2 hours ago
Polynomials
CuriousBabu   10
N Apr 18, 2025 by wh0nix
\[ 
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 
\]
Find the number of real solutions.
10 replies
CuriousBabu
Apr 14, 2025
wh0nix
Apr 18, 2025
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CuriousBabu
10 posts
#1 • 1 Y
Y by PikaPika999
\[ 
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 
\]
Find the number of real solutions.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lbh_qys
563 posts
#3 • 1 Y
Y by PikaPika999
Assuming that $x+y = 2a$, $y+z = 2b$, and $z+x = 2c$, it is easy to prove that the left-hand side of the equation can be reduced to a quadratic polynomial in $a$, $b$, and $c$, and that it is an even function of $a$, $b$, and $c$. Consequently, it must be a constant multiple of $a^2+b^2+c^2$. Since it is evident that the left-hand side is not identically zero, we obtain $a^2+b^2+c^2=0$, which implies that $a=b=c=0$. However, at this point the denominator becomes zero, hence the original equation possesses no real solution.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
41 posts
#5 • 1 Y
Y by PikaPika999
CuriousBabu wrote:
\[ 
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 
\]
Find the number of real solutions.

First, we have $(x+y)(y+z)(z+x) \neq 0$

$LHS=\frac{5(x+y)(y+z)(z+x)(x^2+y^2+z^2+xy+yz+zx)}{(x+y)(y+z)(z+x)}=5(x^2+y^2+z^2+xy+yz+zx)=0$

So $x^2+y^2+z^2+xy+yz+zx=\frac{1}{2}((x+y)^2+(y+z)^2+(z+x)^2))=0 \Leftrightarrow x+y=y+z=z+x=0$.But it's contradiction to $(x+y)(y+z)(z+x) \neq 0$

So this equation doesn't have real solutions.
This post has been edited 1 time. Last edited by lgx57, Apr 18, 2025, 8:44 AM
Reason: Mistake in Latex
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wh0nix
15 posts
#6 • 1 Y
Y by PikaPika999
How do you get the factorization from the numerator?
This post has been edited 1 time. Last edited by wh0nix, Apr 18, 2025, 9:15 AM
Reason: typo
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Pengu14
609 posts
#7 • 1 Y
Y by PikaPika999
wh0nix wrote:
How do you get the factorization from the numerator?

If you substitute x=-y, the numerator becomes 0. This means that x+y is a factor. Similarly, you can show y+z and x+z also must be factors.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wh0nix
15 posts
#8 • 1 Y
Y by PikaPika999
I thought about that too. I also thought that the quotient should be a quadratic symmetric polynomial. But why this one?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
41 posts
#9 • 1 Y
Y by PikaPika999
wh0nix wrote:
How do you get the factorization from the numerator?

You can see it in https://baike.baidu.com/item/%E8%BD%AE%E6%8D%A2%E5%BC%8F/5656585?fromtitle=%E8%BD%AE%E6%8D%A2%E5%AF%B9%E7%A7%B0%E5%BC%8F&fromid=613623

This website is in Chinese.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lgx57
41 posts
#10 • 1 Y
Y by PikaPika999
wh0nix wrote:
I thought about that too. I also thought that the quotient should be a quadratic symmetric polynomial. But why this one?

There are two basic quadratic symmetric polynomials. They are $ab+bc+ca$ and $a^2+b^2+c^2$. Both of them can be in the formula.
So if you want to show a quadratic symmetric polynomial ,you can use $k_1(a^2+b^2+c^2)+k_2(ab+bc+ca)$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wh0nix
15 posts
#11
Y by
Thank you much! How we determine the constant k1 and k2?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Jackson0423
84 posts
#12
Y by
Put some easy numbers,, ex a=1, b=1, c=0 or a=0, b=1, c=-1 like this
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
wh0nix
15 posts
#13
Y by
Now I understand it! Thank you!
Z K Y
N Quick Reply
G
H
=
a