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

We have your learning goals covered with Spring and Summer courses available. Enroll today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/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, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
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
Tuesday, Mar 25 - Jul 8
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
Sunday, Mar 23 - Jul 20
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
Sunday, Mar 16 - Jun 8
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
Monday, Mar 17 - Jun 9
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
Sunday, Mar 2 - Jun 22
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
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
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
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
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
Sunday, Mar 23 - Aug 3
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
Sunday, Mar 16 - Aug 24
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
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
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
Sunday, Mar 23 - Jun 15
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
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
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
Monday, Mar 24 - Jun 16
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
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
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
Mar 2, 2025
0 replies
J
H
Nesbitt in triangle
Dynic   0
26 minutes ago
Let $a,b,c$ be the lengths of the three sides of a triangle, respectively. Let $\mathbb{X}$ as a set of positive real number and the function $D:\mathbb{X}^3\to\mathbb{X}$ determined by $$D(x,y,z)=\dfrac{x}{y+z}+\dfrac{y}{x+z}+\dfrac{z}{x+y},\forall x,y,z >0$$Find all values that $D(a,b,c)$ can take.
0 replies
Dynic
26 minutes ago
0 replies
J
H
Geometry / Ellipses
SpacePhysics   0
28 minutes ago
From three points A,B,C fixed in space is formed an acute triangle ABC, and three ellipses such that the points A,B,C are their foci, and they all intersect in a point P inside the triangle. Find P such that the sum of the semi-major axis from the three ellipses is minimized
0 replies
SpacePhysics
28 minutes ago
0 replies
J
H
a+b+c=3 ine
jokehim   0
29 minutes ago
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\color{black}{\frac{a\left(b+c\right)}{bc+3}+\frac{b\left(c+a\right)}{ca+3}+\frac{c\left(a+b\right)}{ab+3}\le \frac{3}{2}.}$$Proposed by Phan Ngoc Chau
0 replies
1 viewing
jokehim
29 minutes ago
0 replies
J
H
Inequalities
sqing   3
N an hour ago by sqing
Let $ a,b\geq 2 . $ Prove that
$$ (1-a^2)(1-b^2) (ab-1)^2-108 ab \geq-351 $$$$(1-a^2)(1-b^2) (ab-1)(a^2b^2-1)- 621 ab \geq-2079 $$
3 replies
sqing
an hour ago
sqing
an hour ago
J
H
Differentiation Marathon!
LawofCosine   185
N Today at 12:24 AM by LawofCosine
Hello, everybody!

This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!

Have fun!

(Sorry about the bad english)
185 replies
LawofCosine
Feb 1, 2025
LawofCosine
Today at 12:24 AM
J
H
Integration Bee Kaizo
Calcul8er   46
N Today at 12:18 AM by awzhang10
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
46 replies
Calcul8er
Mar 2, 2025
awzhang10
Today at 12:18 AM
J
H
Geometric Optimization Problem
ReticulatedPython   0
Yesterday at 6:59 PM
Source: Myself
Consider three concentric circles with radii of lengths $a$, $b$, and $c$, with $a<b<c.$ Point $A$ is chosen on the circle with radius $a$, point $B$ is chosen on the circle with radius $b$, and point $C$ is chosen on the circle with radius $c.$ Find (in terms of $a$, $b$, and $c$):

(a) The maximum possible area of $\triangle{ABC}.$
(b)The maximum possible perimeter of $\triangle{ABC}.$
0 replies
ReticulatedPython
Yesterday at 6:59 PM
0 replies
J
H
f must be a constant function
WakeUp   2
N Yesterday at 3:31 PM by Fibonacci_math
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that
\[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\]
Prove that $f$ is a constant function.
2 replies
WakeUp
Dec 8, 2010
Fibonacci_math
Yesterday at 3:31 PM
J
H
real analysis
ay19bme   1
N Yesterday at 3:21 PM by Etkan
Show that $f(x)=\left(1-\dfrac{1}{x}\right)^x$ is increasing for all $x\in[1,\infty)$.
1 reply
ay19bme
Yesterday at 2:31 PM
Etkan
Yesterday at 3:21 PM
J
H
probability prime numbers
danilorj   0
Yesterday at 1:28 PM
What is the probability of the sum of the numbers showing up on the top of 100 six-sided dice being a prime number when they are thrown simultaneously?
0 replies
danilorj
Yesterday at 1:28 PM
0 replies
J
H
Concavity of a function
pii-oner   2
N Yesterday at 10:12 AM by pii-oner
Hi everyone,

I am studying the concavity of the function

\[ f(x) = \sqrt{1 - x^a}, \quad a \geq 0 \]
on the interval \( x \in [0,1] \).

I computed the second derivative and found that for \( a \geq 1 \), the function appears to be concave. However, I am uncertain about the behavior at the endpoints.

Does anyone have insights on confirming concavity rigorously for \( a \geq 1 \) and understanding the behavior at the endpoints? Any help would be greatly appreciated!

Thanks!
2 replies
pii-oner
Mar 16, 2025
pii-oner
Yesterday at 10:12 AM
J
H
Do these have a closed form?
Entrepreneur   0
Yesterday at 8:18 AM
Source: Own
$$\int_0^\infty\frac{t^{n-1}}{(t+\alpha)^2+m^2}dt.$$$$\int_0^\infty\frac{e^{nt}}{(t+\alpha)^2+m^2}dt.$$$$\int_0^\infty\frac{dx}{(1+x^a)^m(1+x^b)^n}.$$
0 replies
Entrepreneur
Yesterday at 8:18 AM
0 replies
J
H
Interesting limit with bijective function
AndreiVila   3
N Yesterday at 3:46 AM by Levieee
Source: Romanian District Olympiad 2025 11.3
Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$[list=a]
[*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$.
[*] Give an example of function which satisfies the hypothesis.
3 replies
AndreiVila
Mar 8, 2025
Levieee
Yesterday at 3:46 AM
J
H
Double factorial identity
Snoop76   0
Sunday at 6:31 PM
Source: own
Show that the following identity holds:$$\sum_{k=0}^n (2k+3)!!{n\choose k}=2(n+1)\sum_{k=0}^n (2k+1)!!{n\choose k}+\sum_{k=0}^n (2k-1)!!{n\choose k}$$
0 replies
Snoop76
Sunday at 6:31 PM
0 replies
J
H
J
H
Classic geometry problem
Raul_S_Baz   2
N Yesterday at 4:10 PM by Raul_S_Baz
IMAGE
2 replies
Raul_S_Baz
Mar 4, 2025
Raul_S_Baz
Yesterday at 4:10 PM
J
Classic geometry problem
G H J
Reveal topic tags
geometry
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.
Raul_S_Baz
9 posts
Raul_S_Baz
#1 h  Mar 4, 2025, 3:49 AM
Y by
https://imgur.com/a/Oi4SbIj
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sunken rock
4372 posts
sunken rock
#2 h  Mar 4, 2025, 11:42 AM • 2 Y
Y by Raul_S_Baz, sangriabeaver5
$\triangle ADC\sim\triangle BNC$; as $AD=2CD$, it follows that $BN=2CN$, done.

Best regards,
sunken rock
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Raul_S_Baz
9 posts
Raul_S_Baz
#3 h  Yesterday at 4:10 PM
Y by
Thanx a lot! Very fine solution!
Z K Y
N Quick Reply
G
H
=
a