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
n Tans
MithsApprentice   17
N 2 minutes ago by AshAuktober
Source: USAMO 1998
Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[ \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. \] Prove that \[ \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}. \]
17 replies
+1 w
MithsApprentice
Oct 9, 2005
AshAuktober
2 minutes ago
J
H
Roots of unity problem
Ferid.---.   10
N 12 minutes ago by Avron
Source: Polish MO 2019 P4
Let $n,k,l$ be positive integers.Define injective function $f$ from $\{1,2,\dots,n\}$ to itself such that $f(i)-i\in \{k,-l\}$.Prove that $k+l$ divides $n$.
10 replies
Ferid.---.
May 19, 2019
Avron
12 minutes ago
J
H
Coloring
demmy   6
N 23 minutes ago by Kaimiaku
Source: Thailand TST 2015
What is the maximum number of squares in an $8 \times 8$ board that can be colored so that for each square in the board, at most one square adjacent to it is colored.
6 replies
demmy
Dec 2, 2023
Kaimiaku
23 minutes ago
J
H
Nice and easy FE on R+
sttsmet   21
N 28 minutes ago by bo18
Source: EMC 2024 Problem 4, Seniors
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
21 replies
sttsmet
Dec 23, 2024
bo18
28 minutes ago
J
H
Differentiation Marathon!
LawofCosine   181
N Today at 12:24 AM by Levieee
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)
181 replies
LawofCosine
Feb 1, 2025
Levieee
Today at 12:24 AM
J
H
Integrals problems and inequality
tkd23112006   2
N Yesterday at 6:02 PM by PolyaPal
Let f be a continuous function on [0,1] such that f(x) ≥ 0 for all x ∈[0,1] and
$\int_x^1 f(t) dt \geq \frac{1-x^2}{2}$ , ∀x∈[0,1].
Prove that:
$\int_0^1 (f(x))^{2021} dx \geq \int_0^1 x^{2020} f(x) dx$
2 replies
tkd23112006
Feb 16, 2025
PolyaPal
Yesterday at 6:02 PM
J
H
real analysis
ay19bme   1
N Yesterday at 5:30 PM by alexheinis
.........
1 reply
ay19bme
Yesterday at 3:05 PM
alexheinis
Yesterday at 5:30 PM
J
H
Proving an inequality involving cosine functions
pii-oner   3
N Yesterday at 3:33 PM by pii-oner
Hello AoPS Community,

I am curious about how to demonstrate the following inequality:

[code] \sqrt{1 - |\cos(x \pm y)|^a} \leq \sqrt{1 - |\cos(x)|^a} + \sqrt{1 - |\cos(y)|^a}, \quad \text{for } a \geq 1. [/code]

I’ve plotted the functions, and the inequality seems to hold. However, I am looking for a rigorous way to prove it.

I’d greatly appreciate insights into how to break this down analytically and references to similar problems or techniques that might be helpful.

Thank you so much for your guidance and support!
3 replies
pii-oner
Jan 22, 2025
pii-oner
Yesterday at 3:33 PM
J
H
Linear algebra
Dynic   2
N Yesterday at 3:32 PM by loup blanc
Let A and B be two square matrices with the same size. Prove that if AB is an invertible matrix, then A and B are also invertible matrices
2 replies
Dynic
Yesterday at 2:48 PM
loup blanc
Yesterday at 3:32 PM
J
H
Very hard group theory problem
mathscrazy   3
N Yesterday at 9:50 AM by quasar_lord
Source: STEMS 2025 Category C6
Let $G$ be a finite abelian group. There is a magic box $T$. At any point, an element of $G$ may be added to the box and all elements belonging to the subgroup (of $G$) generated by the elements currently inside $T$ are moved from outside $T$ to inside (unless they are already inside). Initially $ T$ contains only the group identity, $1_G$. Alice and Bob take turns moving an element from outside $T$ to inside it. Alice moves first. Whoever cannot make a move loses. Find all $G$ for which Bob has a winning strategy.
3 replies
mathscrazy
Dec 29, 2024
quasar_lord
Yesterday at 9:50 AM
J
H
limit of u(pi/45)
EthanWYX2009   0
Yesterday at 7:08 AM
Source: 2025 Pi Day Challenge T5
Let \(\omega\) be a positive real number. Divide the positive real axis into intervals \([0, \omega)\), \([\omega, 2\omega)\), \([2\omega, 3\omega)\), \([3\omega, 4\omega)\), \(\ldots\), and color them alternately black and white. Consider the function \(u(x)\) satisfying the following differential equations:
\[ u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals}, \]\[ u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals}, \]with the initial conditions:
\[ u(0) = 1, \quad u'(0) = 1, \]and the continuity conditions:
\[ u(x) \text{ and } u'(x) \text{ are continuous functions}. \]Show that
\[ \lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0. \]
0 replies
EthanWYX2009
Yesterday at 7:08 AM
0 replies
J
H
Integration Bee Kaizo
Calcul8er   42
N Yesterday at 12:57 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!
42 replies
Calcul8er
Mar 2, 2025
awzhang10
Yesterday at 12:57 AM
J
H
maximum value
Tip_pay   3
N Friday at 9:37 PM by alexheinis
Find the value $x\in [0,4]$ at which the function $f(x)=\int_{0}^{\sqrt{x}}\ln \frac{e}{1+t^2}dt$ takes its maximum value
3 replies
Tip_pay
Friday at 8:48 PM
alexheinis
Friday at 9:37 PM
J
H
Spheres and a point source of light
mofidy   3
N Friday at 9:22 PM by kiyoras_2001
How many spheres are needed to shield a point source of light?
Unfortunately, I didn't find a suitable solution for it on the page below:
https://artofproblemsolving.com/community/c4h1469498p8521602
Here too, two different solutions are given:
https://math.stackexchange.com/questions/2791186
3 replies
mofidy
Mar 11, 2025
kiyoras_2001
Friday at 9:22 PM
J
H
J
H
geometry
srnjbr   2
N Yesterday at 10:17 PM by maromex
the points f,n,o, t a lie in the plane such that the triangles tfo ton are similar, preserving direction and order, and fano is a parallelogram. show that of×on=oa×ot.
2 replies
srnjbr
Yesterday at 8:16 PM
maromex
Yesterday at 10:17 PM
J
geometry
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.
srnjbr
46 posts
srnjbr
#1 h  Yesterday at 8:16 PM
Y by
the points f,n,o, t a lie in the plane such that the triangles tfo ton are similar, preserving direction and order, and fano is a parallelogram. show that of×on=oa×ot.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
removablesingularity
552 posts
removablesingularity
#2 h  Yesterday at 9:32 PM
Y by
the points are $f,n,o,t$ and $fano$ is a parallelogram, what is $a$?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
maromex
101 posts
maromex
#3 h  Yesterday at 10:17 PM
Y by
$a$ is also a point but it's confusing because there was no comma
Z K Y
N Quick Reply
G
H
=
a