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 and Fall classes are open for enrollment. Schedule today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC 10 Problem Series[/list]
For those interested in Olympiad level training in math, computer science, physics, and chemistry, be sure to enroll in our WOOT courses before August 19th to take advantage of early bird pricing!

Summer camps are starting this 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 a transformative 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][*]June 5th, Thursday, 7:30pm ET: Open Discussion with Ben Kornell and Andrew Sutherland, Art of Problem Solving's incoming CEO Ben Kornell and CPO Andrew Sutherland host an Ask Me Anything-style chat. Come ask your questions and get to know our incoming CEO & CPO!
[*]June 9th, Monday, 7:30pm ET, Game Jam: Operation Shuffle!, Come join us to play our second round of Operation Shuffle! If you enjoy number sense, logic, and a healthy dose of luck, this is the game for you. No specific math background is required; all are welcome.[/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, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

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
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Jun 22 - Nov 2
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3
Wednesday, Sep 24 - Dec 17

Precalculus
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Wednesday, Jun 25 - Dec 17
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
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!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
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!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Monday, Jun 30 - Jul 21
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Sunday, Jun 22 - Sep 21
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Wednesday, Jun 11 - Aug 27
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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
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
Thursday, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Sunday, Jun 22 - Sep 1
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Monday, Jun 23 - Dec 15
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Jun 2, 2025
0 replies
J
H
[Self Made Problem]
reilynso   3
N an hour ago by Rabbit47
Let a, b, and c be the roots of the equation x^3 - 2x^2 + 3x - 4 = 0.

Find 1/a² + 1/b² + 1/c².
3 replies
reilynso
4 hours ago
Rabbit47
an hour ago
J
H
[PMO22 Areas/II.1] Expected value of maximum
tapilyoca   3
N 2 hours ago by steve4916
Consider all subsets of $\{1,2,3,\dots,2018,2019\}$ containing exactly $100$ elements. From each of these subsets, take the greatest element. Find the average of all of these greatest elements.

Answer
3 replies
tapilyoca
Yesterday at 3:14 PM
steve4916
2 hours ago
J
H
Domain of Nested Functions
pensive   1
N 2 hours ago by pensive
Given $f(x) = \log_2 (x) - 1$ and $g(x) = -x^2+x+2$, for what integer $x$ is $f(g(f(x)))$ positive?
Answer
1 reply
pensive
3 hours ago
pensive
2 hours ago
J
H
Algebra equation
Math.hunter   0
3 hours ago
Solve for real x, y.
(x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) = 0
0 replies
Math.hunter
3 hours ago
0 replies
J
H
inequality
luckvoltia.112   0
3 hours ago
$ \text{Let } a, b, c \text{ be non-negative real numbers such that } ab + bc + ca = 1. \text{ Prove that } \frac{(1 + a^2)^2}{b^2 + c^2} + \frac{(1 + b^2)^2}{c^2 + a^2} + \frac{(1 + c^2)^2}{a^2 + b^2} \geq 8. $
0 replies
luckvoltia.112
3 hours ago
0 replies
J
H
Quadratic Equation
EntangledElectron99   0
4 hours ago
If $\alpha \ and \ \beta$ are roots of $ x^2 - 6x +2 = 0$ and $ S(n_1,n_2) = 3\alpha^{n_1} + 5\beta^{n_2} $ for
$ n_1 , n_2 \ge 2 $, then which of the following is true?[Multiple options maybe correct]

a) $ S(100,99) - 6S(99,98) + 2S(98,97) = 0 $
b)$ S(10,11) - 6S(9,10) + 2S(8,9) = 0 $
c) $ S(99,100) - 6S(98,99) + 2S(97,98) = 0 $
d) $ S(11,10) - 6S(10,9) + 2S(9,8) = 0 $

0 replies
EntangledElectron99
4 hours ago
0 replies
J
H
[Sipnayan SHS 2025] Semifinals Round, Average, #7:
LilKirb   1
N 4 hours ago by Viliciri
Let $n$ be a randomly chosen divisor of $30^6.$ Find the probability that $2n$ has more positive divisors than both $3n$ and $5n.$
1 reply
LilKirb
4 hours ago
Viliciri
4 hours ago
J
H
Inequalities
sqing   16
N 4 hours ago by sqing
Let $ a,b> 0 ,\frac{a}{2b+1}+\frac{b}{3}+\frac{1}{2a+1} \leq 1.$ Prove that
$$ a^2+b^2 -ab\leq 1$$$$ a^2+b^2 +ab \leq3$$Let $ a,b,c> 0 , \frac{a}{2b+1}+\frac{b}{2c+1}+\frac{c}{2a+1} \leq 1.$ Prove that
$$ a +b +c +abc \leq 4$$
16 replies
sqing
May 24, 2025
sqing
4 hours ago
J
H
[12th PMO Qualifying Stage for Regions 1, 2, and CAR: II.19]
reilynso   1
N 4 hours ago by reilynso
Three of the roots of the equation 2x^4 + ax^2 + bx + c = 0 are -1, 2, and -3. What is b?

a) 0 b) 6 c) 8 d) -18
1 reply
reilynso
4 hours ago
reilynso
4 hours ago
J
H
[23rd PMO Qualifying Stage: I.5]
reilynso   1
N 4 hours ago by reilynso
Find the sum of all k for which x^5 + kx^4 - 6x^3 - 15x^2 - 8k^3x - 12k + 21 leaves a remainder of 23 when divided by 0.

(a) -1 (b) -3/4 (c) 5/8 (d) 3/4
1 reply
reilynso
4 hours ago
reilynso
4 hours ago
J
H
inequality
luckvoltia.112   4
N 4 hours ago by sqing
Given that $a, b, c$ are non-negative real numbers such that $a + b + c = 3$. Find the maximum value of

$$ P = \sqrt{(a^2 + 3)(b^2 + 3)} + \sqrt{(b^2 + 3)(c^2 + 3)} + \sqrt{(c^2 + 3)(a^2 + 3)}. $$
4 replies
luckvoltia.112
Yesterday at 3:11 PM
sqing
4 hours ago
J
H
Bounding With Powers
Shreyasharma   6
N 5 hours ago by jacosheebay
Is this a valid solution for the following problem (St. Petersburg 1996):

Find all positive integers $n$ such that,

$$ 3^{n-1} + 5^{n-1} | 3^n + 5^n$$
Solution
6 replies
Shreyasharma
Jul 11, 2023
jacosheebay
5 hours ago
J
H
Difficult Triangle Angle Chasing
Viliciri   2
N 5 hours ago by Viliciri
In $\triangle{ABC}$, $\angle{ABC} = 54^{\circ}$ and $\angle{ACB} = 24^{\circ}$.
If $D$ lies on segment $BC$ such that $AC=BD$, what is the measure of $\angle{DAC}$?

Credit to Nathan Liu for communicating this problem to me.
2 replies
Viliciri
Today at 3:12 AM
Viliciri
5 hours ago
J
H
Inequalities
sqing   5
N Today at 11:43 AM by sqing
Let $ a,b,c\geq 0,a+b+c=1. $ Prove that
$$ 7\geq \frac{2+a}{2-a}+\frac{3+2b}{3 -2b}+\frac{2+c}{2 -c}\geq \frac{11+16\sqrt {3}}{9}$$$$5\geq \frac{2+a}{2-a}+\frac{5+2b}{5 -2b}+\frac{2+c}{2 -c}\geq \frac{9+16\sqrt {5}}{11}$$$$5\geq \frac{2+a}{2-a}+\frac{7+2b}{7 -2b}+\frac{2+c}{2 -c}\geq \frac{7+16\sqrt{7}}{13}$$
5 replies
sqing
Jun 13, 2025
sqing
Today at 11:43 AM
J
H
J
H
Counting problem
lgx57   1
N Jun 6, 2025 by HAL9000sk
Calculate the number of $n$ that meet the following conditions:
1. $585 \mid n$
2.$0 \sim 7$ appears exactly once in each octal digit of $n$
1 reply
lgx57
Jun 6, 2025
HAL9000sk
Jun 6, 2025
J
Counting problem
G H J
Reveal topic tags
algebra
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.
lgx57
60 posts
lgx57
#1 h  Jun 6, 2025, 8:36 AM
Y by
Calculate the number of $n$ that meet the following conditions:
1. $585 \mid n$
2.$0 \sim 7$ appears exactly once in each octal digit of $n$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HAL9000sk
60 posts
HAL9000sk
#2 h  Jun 6, 2025, 10:50 AM
Y by
solution
Z K Y
N Quick Reply
G
H
=
a