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 April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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 events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/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, Apr 13 - Aug 10
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
Sunday, Apr 13 - Aug 10
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
Monday, Apr 7 - Jul 28
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
Wednesday, Apr 16 - Jul 2
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
Thursday, Apr 17 - Jul 3
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
Wednesday, Apr 16 - Jul 30
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Wednesday, Apr 23 - Oct 1
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

Intermediate: Grades 8-12

Intermediate Algebra
Monday, Apr 21 - Oct 13
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
Friday, Apr 11 - Jun 27
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Wednesday, Apr 9 - Sep 3
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
Wednesday, Apr 16 - Jul 2
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
Friday, Apr 11 - Jun 27
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

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
Sat & Sun, Apr 26 - Apr 27 (4:00 - 7:00 pm ET/1:00 - 4:00pm PT)
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
>=512 different isosceles triangles whose vertices have the same color
parmenides51   2
N 6 minutes ago by cooljoseph
Source: Mathematics Regional Olympiad of Mexico West 2016 P6
The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.
2 replies
+1 w
parmenides51
Sep 7, 2022
cooljoseph
6 minutes ago
J
H
Fourth power ineq
Project_Donkey_into_M4   1
N 2 hours ago by sqing
Source: 2018 Mock RMO tdp and kayak P1
Let $a,b,c,d \in \mathbb{R}^+$ such that $a+b+c+d \leq 1$. Prove that\[\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq 255\cdot abcd.\]
1 reply
Project_Donkey_into_M4
Yesterday at 6:20 PM
sqing
2 hours ago
J
H
Is this FE solvable?
ItzsleepyXD   0
2 hours ago
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
0 replies
ItzsleepyXD
2 hours ago
0 replies
J
H
Dear Sqing: So Many Inequalities...
hashtagmath   36
N 2 hours ago by sqing
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
36 replies
hashtagmath
Oct 30, 2024
sqing
2 hours ago
J
H
In a school of $800$ students, $224$ students play cricket, $240$ students play
Vulch   1
N 4 hours ago by RollingPanda4616
Hello everyone,
In a school of $800$ students, $224$ students play cricket, $240$ students play hockey and $336$ students play basketball. $64$ students play both basketball and hockey, $80$ students play both cricket and basketball, $40$ students play both cricket and hockey, and $24$ students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.

Edit:
In the above problem,I just want to know that why the number of students who don't play any game shouldn't be 0, because,if we add 224,240 and 336 it comes out to be 800.I have solution,but I just want to know how to explain it without theoretically.Thank you!
1 reply
Vulch
5 hours ago
RollingPanda4616
4 hours ago
J
H
100th post
MathJedi108   1
N 5 hours ago by mdk2013
Well I guess this is my 100th post, it would be really funny if it isn't can yall share your favorite experience on AoPS here?
1 reply
MathJedi108
6 hours ago
mdk2013
5 hours ago
J
H
Find all triples
pedronis   2
N 6 hours ago by Kempu33334
Find all triples of positive integers $(n, r, s)$ such that $n^2 + n + 1$ divides $n^r + n^s + 1$.
2 replies
pedronis
Apr 19, 2025
Kempu33334
6 hours ago
J
H
Median geometry
Sedro   4
N Yesterday at 10:01 PM by Sedro
In triangle $ABC$, points $D$, $E$, and $F$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Prove that the area of the triangle with side lengths $AD$, $BE$, and $CF$ has area $\tfrac{3}{4}[ABC]$.
4 replies
Sedro
Yesterday at 6:03 PM
Sedro
Yesterday at 10:01 PM
J
H
Combinatorial proof
MathBot101101   5
N Yesterday at 9:56 PM by Kempu33334
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.
5 replies
MathBot101101
Yesterday at 7:37 AM
Kempu33334
Yesterday at 9:56 PM
J
H
geometry
carvaan   1
N Yesterday at 6:38 PM by Lankou
The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.
1 reply
carvaan
Yesterday at 5:46 PM
Lankou
Yesterday at 6:38 PM
J
H
weird permutation problem
Sedro   1
N Yesterday at 6:07 PM by Sedro
Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$. How many possible $\sigma$ exist?
1 reply
Sedro
Yesterday at 2:09 AM
Sedro
Yesterday at 6:07 PM
J
H
Recursion
Sid-darth-vater   6
N Yesterday at 5:59 PM by vanstraelen
Help, I can't characterize ts and I dunno what to do
6 replies
Sid-darth-vater
Yesterday at 3:02 AM
vanstraelen
Yesterday at 5:59 PM
J
H
geometry
carvaan   0
Yesterday at 5:48 PM
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
0 replies
carvaan
Yesterday at 5:48 PM
0 replies
J
H
Graph of polynomials
Ecrin_eren   1
N Yesterday at 5:36 PM by vanstraelen
The graph of the quadratic polynomial with real coefficients y = px^2 + qx + r, called G1, intersects the graph of the polynomial y = x^2, called G2, at points A and B. The lines tangent to G2 at points A and B intersect at point C. It is known that point C lies on G1. What is the value of p?
1 reply
Ecrin_eren
Yesterday at 3:00 PM
vanstraelen
Yesterday at 5:36 PM
J
H
J
H
A Generalization of Ptolemy's Theorem
buratinogigle   0
Apr 16, 2025
Source: VN Math Olympiad For High School Students P7 - 2025
Given a convex quadrilateral $ABCD$, define
\[ \alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC| \quad\text{and}\quad \beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|. \]Prove that
\[ AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta. \]
0 replies
buratinogigle
Apr 16, 2025
0 replies
J
A Generalization of Ptolemy's Theorem
G H J
Reveal topic tags
geometry
L
G H BBookmark kLocked kLocked NReply
Source: VN Math Olympiad For High School Students P7 - 2025
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
buratinogigle
2344 posts
buratinogigle
#1 h  Apr 16, 2025, 1:35 AM
Y by
Given a convex quadrilateral $ABCD$, define
\[ \alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC| \quad\text{and}\quad \beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|. \]Prove that
\[ AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta. \]
Z K Y
N Quick Reply
G
H
=
a