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

Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!
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
Find the number of ordered triples (p,q,r)
Darealzolt   2
N an hour ago by elizhang101412
Let \(p,q,r\) be prime numbers such that
\[ \frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839} \]Hence find the numbers of ordered triples \(\{p,q,r\}\)
2 replies
Darealzolt
2 hours ago
elizhang101412
an hour ago
J
H
Find the sum of all the products a_i a_j
Darealzolt   1
N an hour ago by alexheinis
Among the 100 constants \( \{ a_1,a_2,a_3,\dots,a_{100} \} \),there are \(39\) equal to \( -1\), and \(61\) equal to \(1\). Find the sum of all the products \(a_i a_j\) , where \(a \leq i < j \leq 100\).
1 reply
Darealzolt
4 hours ago
alexheinis
an hour ago
J
H
IOQM P26 2024
SomeonecoolLovesMaths   5
N 2 hours ago by SomeonecoolLovesMaths
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
5 replies
SomeonecoolLovesMaths
Sep 8, 2024
SomeonecoolLovesMaths
2 hours ago
J
H
IOQM 2023-24 P-4
lifeismathematics   8
N 3 hours ago by SomeonecoolLovesMaths
Let $x, y$ be positive integers such that
$$ x^4=(x-1)\left(y^3-23\right)-1 . $$Find the maximum possible value of $x+y$.
8 replies
lifeismathematics
Sep 3, 2023
SomeonecoolLovesMaths
3 hours ago
J
H
Blog Post 51
EpicSkills32   1
N Feb 2, 2014 by briantix
$\ [\text{Blog Post 51}] $

So a few blog posts ago, I posted a problem.

Find all solutions to the following:
\[ x^4-4x=1 \]

Oh dear. We have something to the fourth power. What's even worse, there isn't anything cubed or squared. hm....

At first, we see the 1 and the even power, and we do something like:

\[ x^4-1=4x \]
\[ (x^2+1)(x^2-1)=4x \]
\[ (x^2+1)(x+1)(x-1)=4x \]

And then we get stuck.
Stumped?

Instead, let's try to force a nicer polynomial.
We add $\ 2x^2+1 $ to both sides.

\[ x^4+2x^2-4x+1=2x^2+2 \]
\[ x^4+2x^2+1=2x^2+4x+2 \]

Aha! Something that can be factored!

\[ (x^2+1)^2=2(x^2+2x+1) \]
\[ (x^2+1)^2=2(x+1)^2 \]

Oh but wait a second. The term on the left can be changed so that there is no coefficient in front.

\[ 2(x+1)^2 = 2x^2+4x+2 = (\sqrt{2}x+\sqrt{2})^2 \]

Now we can express our equation with two squared terms:

\[ (x^2+1)^2=(\sqrt{2}x+\sqrt{2})^2 \]

Since both sides are squared, we move both terms to one side to get a difference of squares:

\[ (x^2+1)^2-(\sqrt{2}x+\sqrt{2})^2 \]

Expanding:

\[ \left((x^2+1)+(\sqrt{2}x+\sqrt{2})\right)\left((x^2+1)-(\sqrt{2}x+\sqrt2})\right)= 0 \]

\[ (x^2+\sqrt{2}x+\sqrt{2}+1)(x^2-\sqrt{2}-\sqrt{2}+1) = 0 \]

We have two factors that we need to set equal to zero. Since they are just quadratics, we can do so with the quadratic formula:

\[ (x^2+\sqrt{2}x+\sqrt{2}+1)=0 \]

\[ a= 1 \] \[b=\sqrt{2}\] \[c=\sqrt{2}+1\]

\[ \therefore x= \dfrac{-\sqrt{2}\pm\sqrt{2-4(\sqrt{2}+1)}}{2} \]
\[ x= \dfrac{-\sqrt{2}\pm\sqrt{-2-4\sqrt{2}}}{2} \]

We have these ugly negatives under a terrible radical, so we change that:

\[ x= \dfrac{-\sqrt{2}\pm i\sqrt{2+4\sqrt{2}}}{2} \]

Again for the other factor:

\[ (x^2-\sqrt{2}-\sqrt{2}+1)=0 \]

\[ a=1 \] \[b=-\sqrt{2}\] \[c=-\sqrt{2}+1 \]

\[ \therefore x= \dfrac{\sqrt{2}\pm\sqrt{2-4(-\sqrt{2}+1)}}{2} \]
\[ x= \dfrac{-\sqrt{2}\pm\sqrt{-2+4\sqrt{2}}}{2} \]

In the end, our solutions for $\ x^4-4x=1 $ are the following values:

\[ x= \left(\dfrac{-\sqrt{2}\pm i\sqrt{2+4\sqrt{2}}}{2}\right),\left(\dfrac{-\sqrt{2}\pm\sqrt{-2+4\sqrt{2}}}{2}\right) \]

Yay. :)
1 reply
EpicSkills32
Jan 24, 2014
briantix
Feb 2, 2014
J
No more topics!
H
J
H
Find all tuples of real numbers
nhathhuyyp5c   0
Apr 13, 2025
Let \( n \geq 2 \) be an integer. Find all tuples of real numbers \( (a_1, a_2, \ldots, a_n) \) such that
\[ 2a_1 - 3a_2,\ 2a_2 - 3a_3,\ \ldots,\ 2a_n - 3a_1 \]is a permutation of \( a_1, a_2, \ldots, a_n \).
0 replies
nhathhuyyp5c
Apr 13, 2025
0 replies
J
Find all tuples of real numbers
G H J
Reveal topic tags
number theory
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.
nhathhuyyp5c
63 posts
nhathhuyyp5c
#1 h  Apr 13, 2025, 8:03 AM
Y by
Let \( n \geq 2 \) be an integer. Find all tuples of real numbers \( (a_1, a_2, \ldots, a_n) \) such that
\[ 2a_1 - 3a_2,\ 2a_2 - 3a_3,\ \ldots,\ 2a_n - 3a_1 \]is a permutation of \( a_1, a_2, \ldots, a_n \).
Z K Y
N Quick Reply
G
H
=
a