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 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
J
H
36x⁴ + 12x² - 36x + 13 > 0
fxandi   1
N an hour ago by idk12345678
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
1 reply
fxandi
3 hours ago
idk12345678
an hour ago
J
H
trigonometric functions
VivaanKam   12
N 2 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
12 replies
VivaanKam
Apr 29, 2025
aok
2 hours ago
J
H
2025 OMOUS Problem 4
enter16180   2
N 4 hours ago by Acridian9
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Find all matrices $M \in M_{n}(\mathbb{C})$ such that following equality holds

$$ \operatorname{rank}(M)+\operatorname{rank}\left(M^{2023}-M^{2025}\right)=\operatorname{rank}\left(M-M^{2}\right)+\operatorname{rank}\left(M^{2023}+M^{2024}\right) $$
2 replies
enter16180
Apr 18, 2025
Acridian9
4 hours ago
J
H
Determinant is 1
Entrepreneur   1
N 5 hours ago by Etkan
If a determinant is of $n^{\text{th}}$ order, and if the constituents of its first, second, ..., $n^{\text{th}}$ rows are the first $n$ figurate numbers of the first, second, ..., $n^{\text{th}}$ orders respectively, show that it's value is $1.$
1 reply
Entrepreneur
Yesterday at 7:14 PM
Etkan
5 hours ago
J
H
A Collection of Good Problems from my end
SomeonecoolLovesMaths   8
N 5 hours ago by Math-lover1
This is a collection of good problems and my respective attempts to solve them. I would like to encourage everyone to post their solutions to these problems, if any. This will not only help others verify theirs but also perhaps bring forward a different approach to the problem. I will constantly try to update the pool of questions.

The difficulty level of these questions vary from AMC 10 to AIME. (Although the main pool of questions were prepared as a mock test for IOQM over the years)

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5
8 replies
SomeonecoolLovesMaths
Sunday at 8:16 AM
Math-lover1
5 hours ago
J
H
find number of elements in H
Darealzolt   1
N Yesterday at 6:47 PM by alexheinis
If \( H \) is the set of positive real solutions to the system
\[ x^3 + y^3 + z^3 = x + y + z \]\[ x^2 + y^2 + z^2 = xyz \]then find the number of elements in \( H \).
1 reply
Darealzolt
Yesterday at 1:50 AM
alexheinis
Yesterday at 6:47 PM
J
H
Poker hand
Aksudon   1
N Yesterday at 6:32 PM by lucaminiati
Problem: In a standard 52-card deck, how many different five-card poker hands are there of 'two pairs'?

Can someone please explain what is logically wrong with the following solution? (It gives double of the right solution which supposed to be 123552).

13\binom{4}{2}*12\binom{4}{2}*44=247104

Thanks
1 reply
Aksudon
Yesterday at 5:14 PM
lucaminiati
Yesterday at 6:32 PM
J
H
primes and perfect squares
Bummer12345   0
Yesterday at 5:08 PM
If $p$ and $q$ are primes, then can $2^p + 5^q + pq$ be a perfect square?
0 replies
Bummer12345
Yesterday at 5:08 PM
0 replies
J
H
simple trapezoid
gggzul   0
Yesterday at 4:44 PM
Let $ABCD$ be a trapezoid. By $x$ we denote the angle bisector of angle $X$ . Let $P=a\cap c$ and $Q=b\cap d$. Prove that $ABPQ$ is cyclic.
0 replies
gggzul
Yesterday at 4:44 PM
0 replies
J
H
geometry
JetFire008   0
Yesterday at 4:14 PM
Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.
0 replies
JetFire008
Yesterday at 4:14 PM
0 replies
J
H
Inequalities
sqing   11
N Yesterday at 3:02 PM by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
11 replies
sqing
Jul 12, 2024
sqing
Yesterday at 3:02 PM
J
H
A rather difficult question
BeautifulMath0926   3
N Yesterday at 2:23 PM by evankuang
I got a difficult equation for users to solve:
Find all functions f: R to R, so that to all real numbers x and y,
1+f(x)f(y)=f(x+y)+f(xy)+xy(x+y-2) holds.
3 replies
BeautifulMath0926
Apr 13, 2025
evankuang
Yesterday at 2:23 PM
J
H
The return of an inequality
giangtruong13   4
N Yesterday at 1:26 PM by sqing
Let $a,b,c$ be real positive number satisfy that: $a+b+c=1$. Prove that: $$\sum_{cyc} \frac{a}{b^2+c^2} \geq \frac{3}{2}$$
4 replies
giangtruong13
Mar 18, 2025
sqing
Yesterday at 1:26 PM
J
H
Polynomial
kellyelliee   1
N Yesterday at 1:19 PM by Jackson0423
Let the polynomial $f(x)=x^2+ax+b$, where $a,b$ integers and $k$ is a positive integer. Suppose that the integers
$m,n,p$ satisfy: $f(m), f(n), f(p)$ are divisible by k. Prove that:
$(m-n)(n-p)(p-m)$ is divisible by k
1 reply
kellyelliee
Yesterday at 3:57 AM
Jackson0423
Yesterday at 1:19 PM
J
H
J
H
invertible matrix
jokerjoestar   2
N Mar 31, 2025 by SatisfiedMagma
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
2 replies
jokerjoestar
Mar 27, 2025
SatisfiedMagma
Mar 31, 2025
J
invertible matrix
G H J
Reveal topic tags
linear algebra
matrix
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.
jokerjoestar
150 posts
jokerjoestar
#1 h  Mar 27, 2025, 6:50 AM
Y by
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexheinis
10581 posts
alexheinis
#2 h  Mar 27, 2025, 9:31 AM • 3 Y
Y by jokerjoestar, SatisfiedMagma, MihaiT
The minimal polynomial divides $x^{2025}-2025x=x(x^{2024}-2025)$ hence it is coprime to $x-1$.
Or let's do it explicitly here: if $Ax=x$ then $x=A^{2025}(x)=2025 Ax=2025 x$. Hence $x=0$.
This post has been edited 1 time. Last edited by alexheinis, Mar 27, 2025, 9:46 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SatisfiedMagma
458 posts
SatisfiedMagma
#3 h  Mar 31, 2025, 10:29 AM
Y by
Here's another solution found by my batchmate Amit Prakash Jena.

Solution: Let $A - I = X$, then observe that
\begin{align*} A^{2025} &= (X+I)^{2025} \\ 2025A &= \sum_{i = 0}^{2025}X^i \binom{2025}{i} \\ 2025(X+I) &= X^{2025} + \ldots + 2025X + I \\ \cancel{2025X} + 2024I &= X\underbrace{(X^{2024} + \ldots + \binom{2025}{2}X)}_{\coloneqq B} + \cancel{2025X} \end{align*}This clearly means that $X\cdot \frac{B}{2024} = I$, proving the invertibility of $X$ since everything is a square matrix. $\blacksquare$
Z K Y
N Quick Reply
G
H
=
a