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

Plan ahead for the next school year. Schedule your class today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
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
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
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
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
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
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
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, 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!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
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, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
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

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

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
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
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
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
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
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
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
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
Thursday, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

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

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
J
H
[20th PMO Area Stage: I. 5]
reilynso   1
N 2 hours ago by P0tat0b0y
Let $f(x)=\sqrt{4 \sin^4 x - \sin ^2 x \cos ^2 x + 4 \cos^4 x}$ for any $x \in \mathbb{R}$. Let $M$ and $m$ be the maximum and minimum values of $f$, respectively. Find the product of $M$ and $m$.

Answer

Solution
1 reply
reilynso
3 hours ago
P0tat0b0y
2 hours ago
J
H
[2025 Sipnayan SHS] Trig Identities
aaa12345   2
N 3 hours ago by reilynso
If \tan x = \sqrt{15} and \sin x < 0, find \cos (4x).
Answer
Solution: https://imgur.com/a/33ab181
Source: 2025 Sipnayan SHS Orals Final Round/Easy Lechon
2 replies
aaa12345
Jun 18, 2025
reilynso
3 hours ago
J
H
[2021 Sipnayan JHS] Trig Identities
aaa12345   2
N 3 hours ago by reilynso
Solve for the value of \frac{8xy-16x^3y}{y^2-x^2} given that x=\sin15^{\circ} and y=\cos15^{\circ}.
Answer
Solution: https://imgur.com/a/SnS8WCL
Source: 2021 Sipnayan JHS Orals/Semifinals Average #1
2 replies
aaa12345
Jun 18, 2025
reilynso
3 hours ago
J
H
An Angle Trisector
bryanguo   2
N 3 hours ago by Amkan2022
Triangle $ABC$ has points $D$,$E$,$F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$, and $AD$ and $AE$ trisect angle $BAF$. If $\angle BAF = 60^{\circ}$, $\frac{EF}{EC}=\frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$.

Individual #5
2 replies
bryanguo
Apr 11, 2024
Amkan2022
3 hours ago
J
H
Trigonometry bash
ehz2701   0
3 hours ago
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.)

Problem Set 1a. Show that
$\begin{aligned} & \mathbf{(1)\quad} \sin(18^{\circ})\sin(30^{\circ}) = \sin(12^{\circ})\sin(48^{\circ}) \\ & \mathbf{(2)\quad} \sin(39^{\circ})\sin(51^{\circ}) = \sin(30^{\circ})\sin(78^{\circ}) \\ & \mathbf{(3)\quad} \sin(24^{\circ})\sin(66^{\circ}) = \sin(30^{\circ})\sin(48^{\circ}) \\ & \mathbf{(4)\quad} \sin(18^{\circ})\sin(78^{\circ}) = \sin(24^{\circ})\sin(48^{\circ}) \\ & \mathbf{(5)\quad} \sin(30^{\circ})\sin(36^{\circ}) = \sin(18^{\circ})\sin(72^{\circ}) \\ & \mathbf{(6)\quad} \sin(30^{\circ})\sin(30^{\circ}) = \sin(18^{\circ})\sin(54^{\circ}) \\ & \mathbf{(7)\quad} \sin(12^{\circ})\sin(24^{\circ}) = \sin(6^{\circ})\sin(54^{\circ}) \\ & \mathbf{(8)\quad} \sin(12^{\circ})\sin(84^{\circ}) = \sin(18^{\circ})\sin(42^{\circ}) \\ & \mathbf{(9)\quad} \sin(18^{\circ})\sin(78^{\circ}) = \sin(24^{\circ})\sin(48^{\circ}) \\ & \mathbf{(10)\quad} \sin(18^{\circ})\sin(54^{\circ}) = \sin(15^{\circ})\sin(75^{\circ}) \\ \end{aligned}$

I admit 1a-6 and 1a-10 is a bit easy analytically. However, the point of the exercise is improving the ability of trigonometric identities.

Problem Set 2a.
$\begin{aligned} & \mathbf{(1)\quad} \sin(27^{\circ})\sin(54^{\circ})\sin(81^{\circ}) =\sin(30^{\circ})\sin(51^{\circ})\sin(69^{\circ}) \\ & \mathbf{(2)\quad} \sin(12^{\circ})\sin(39^{\circ})\sin(81^{\circ}) =\sin(15^{\circ})\sin(30^{\circ})\sin(87^{\circ}) \\ & \mathbf{(3)\quad} \sin(12^{\circ})\sin(72^{\circ})\sin(78^{\circ}) =\sin(24^{\circ})\sin(36^{\circ})\sin(54^{\circ}) \\ & \mathbf{(4)\quad} \sin(18^{\circ})\sin(51^{\circ})\sin(69^{\circ}) =\sin(27^{\circ})\sin(30^{\circ})\sin(81^{\circ}) \\ & \mathbf{(5)\quad} \sin(3^{\circ})\sin(54^{\circ})\sin(81^{\circ}) =\sin(12^{\circ})\sin(15^{\circ})\sin(51^{\circ}) \\ & \mathbf{(6)\quad} \sin(57^{\circ})\sin(81^{\circ})\sin(84^{\circ}) =\sin(66^{\circ})\sin(69^{\circ})\sin(75^{\circ}) \\ & \mathbf{(7)\quad} \sin(33^{\circ})\sin(57^{\circ})\sin(78^{\circ}) =\sin(48^{\circ})\sin(48^{\circ})\sin(54^{\circ}) \\ & \mathbf{(8)\quad} \sin(24^{\circ})\sin(54^{\circ})\sin(84^{\circ}) =\sin(30^{\circ})\sin(42^{\circ})\sin(78^{\circ}) \\ & \mathbf{(9)\quad} \sin(12^{\circ})\sin(48^{\circ})\sin(54^{\circ}) =\sin(30^{\circ})\sin(30^{\circ})\sin(30^{\circ}) \\ & \mathbf{(10)\quad} \sin(9^{\circ})\sin(51^{\circ})\sin(63^{\circ}) =\sin(21^{\circ})\sin(24^{\circ})\sin(48^{\circ}) \\ \end{aligned}$

Problem Set 2b
$\begin{aligned} & \mathbf{(1)\quad} \sin(3^{\circ})\sin(78^{\circ})\sin(x) =\sin(6^{\circ})\sin(39^{\circ})\sin(138^{\circ}-x) \\ & \mathbf{(2)\quad} \sin(18^{\circ})\sin(48^{\circ})\sin(x) =\sin(24^{\circ})\sin(33^{\circ})\sin(101^{\circ}-x) \\ & \mathbf{(3)\quad} \sin(30^{\circ})\sin(54^{\circ})\sin(x) =\sin(36^{\circ})\sin(42^{\circ})\sin(150^{\circ}-x) \\ & \mathbf{(4)\quad} \sin(3^{\circ})\sin(72^{\circ})\sin(x) =\sin(12^{\circ})\sin(24^{\circ})\sin(123^{\circ}-x) \\ & \mathbf{(5)\quad} \sin(3^{\circ})\sin(63^{\circ})\sin(x) =\sin(9^{\circ})\sin(30^{\circ})\sin(99^{\circ}-x) \\ & \mathbf{(6)\quad} \sin(3^{\circ})\sin(69^{\circ})\sin(x) =\sin(6^{\circ})\sin(27^{\circ})\sin(159^{\circ}-x) \\ & \mathbf{(7)\quad} \sin(12^{\circ})\sin(51^{\circ})\sin(x) =\sin(21^{\circ})\sin(24^{\circ})\sin(144^{\circ}-x) \\ & \mathbf{(8)\quad} \sin(18^{\circ})\sin(84^{\circ})\sin(x) =\sin(30^{\circ})\sin(42^{\circ})\sin(150^{\circ}-x) \\ & \mathbf{(9)\quad} \sin(12^{\circ})\sin(42^{\circ})\sin(x) =\sin(21^{\circ})\sin(24^{\circ})\sin(147^{\circ}-x) \\ & \mathbf{(10)\quad} \sin(36^{\circ})\sin(54^{\circ})\sin(x) =\sin(39^{\circ})\sin(51^{\circ})\sin(150^{\circ}-x) \\ \end{aligned}$

answers2b
0 replies
ehz2701
3 hours ago
0 replies
J
H
Geometry easy
AlexCenteno2007   2
N Today at 5:03 AM by AlexCenteno2007
In triangle ABC, if angle B=120°, AB=5u and BC=15u. Draw the interior bisector BE. Calculate BE
2 replies
AlexCenteno2007
Yesterday at 10:55 PM
AlexCenteno2007
Today at 5:03 AM
J
H
Removed coreners from a square
duttaditya18   19
N Today at 3:05 AM by Kaprekar6147
Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area removed to the nearest integer.
19 replies
duttaditya18
Aug 11, 2019
Kaprekar6147
Today at 3:05 AM
J
H
area of right triangle A(T) = E/2+I -1 - Chile 2002 L2 P5
parmenides51   2
N Today at 2:09 AM by Squirrel7O
Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.
2 replies
parmenides51
Sep 1, 2022
Squirrel7O
Today at 2:09 AM
J
H
AOPS Textbook problem
Cookie111   11
N Today at 2:06 AM by Not__Infinity
$\frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}} = 3$
What values of x satisfy this equation
11 replies
Cookie111
Yesterday at 3:32 PM
Not__Infinity
Today at 2:06 AM
J
H
[PMO 26 QUALS]
Shinfu   5
N Today at 1:31 AM by mathprodigy2011
An urn contains two white and two black balls. John draw two balls simultaneously from the urn. If the balls are of different colors, he stops. Otherwise, he returns both balls to the urn and then repeats the process. What is the probability that he stops after exactly three draws?
5 replies
Shinfu
Yesterday at 3:15 PM
mathprodigy2011
Today at 1:31 AM
J
H
Find $a, b$ are positive integers such that $a\le2018$ and $a^5+b^7\vdots 2018
Limited1   2
N Today at 1:01 AM by boylearnmath
Let $a,b$ be positive integers such that $a\le2018$ and $$a^5+b^7\vdots 2018$$. Find $a$
2 replies
Limited1
Aug 14, 2022
boylearnmath
Today at 1:01 AM
J
H
Interesting
AlexCenteno2007   0
Yesterday at 11:02 PM
Let A be a point outside circle k with center 0, and let AP be a
tangent from A to K(P Belongs k). Let B denote the foot of the perpendicular from P to line 0A Choose an arbitrary chord CD in K passing through B and let
E be the reflection of D across AO. Prove that A, C and E are collinear
0 replies
AlexCenteno2007
Yesterday at 11:02 PM
0 replies
J
H
2022 kevinmathz Mock AIME #1 log_{2}\sqrt{ n/10 }}<\sqrt{\log_{2}(n/10 )
parmenides51   2
N Yesterday at 7:54 PM by Tetra_scheme
For how many integers $n$ does the inequality$$\log_{2}\sqrt{\frac{n}{10}}<\sqrt{\log_{2}\left(\frac{n}{10}\right)}$$hold?
2 replies
parmenides51
Dec 17, 2023
Tetra_scheme
Yesterday at 7:54 PM
J
H
Special Multiple of 15
4everwise   21
N Yesterday at 7:52 PM by AbhayAttarde01
The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$.
21 replies
4everwise
Dec 1, 2005
AbhayAttarde01
Yesterday at 7:52 PM
J
H
J
H
Maximum value of function (with two variables)
Saucepan_man02   1
N May 22, 2025 by Saucepan_man02
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
1 reply
Saucepan_man02
May 22, 2025
Saucepan_man02
May 22, 2025
J
Maximum value of function (with two variables)
G H J
Reveal topic tags
function
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.
Saucepan_man02
1400 posts
Saucepan_man02
#1 h  May 22, 2025, 1:25 PM
Y by
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
This post has been edited 1 time. Last edited by Saucepan_man02, May 22, 2025, 1:36 PM
Reason: EDIT
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Saucepan_man02
1400 posts
Saucepan_man02
#2 h  May 22, 2025, 1:39 PM
Y by
Sol
This post has been edited 1 time. Last edited by Saucepan_man02, May 22, 2025, 1:40 PM
Reason: EDIT
Z K Y
N Quick Reply
G
H
=
a