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
Inequalities
sqing   0
33 minutes ago
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
0 replies
1 viewing
sqing
33 minutes ago
0 replies
J
H
one intersting
teomihai   3
N 37 minutes ago by hexuhdecimal
Let $n $ it is fixed positiv number.
Find $card\{(x,y) | |x|+|y|<n \}$, where $x,y$ are integer numbers .
3 replies
teomihai
Mar 15, 2025
hexuhdecimal
37 minutes ago
J
H
Discrete problem
ThanhNg   1
N an hour ago by RPCX
Prove that : From any 161 natural numbers, we can always find 81 numbers whose sum is divisible by 81.
1 reply
ThanhNg
Yesterday at 4:23 PM
RPCX
an hour ago
J
H
Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a p
kyotaro   0
3 hours ago
Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a polynomial satisfying
$$f(a)-f(b) \mid a-b$$
0 replies
kyotaro
3 hours ago
0 replies
J
H
Proof Question
MathRook7817   3
N 5 hours ago by Jack_w
Prove or disprove the assertion that when $1/n$ is written in decimal form with n being a natural number, that it's decimal either terminates or is eventually periodic.

I saw this question somewhere and I thinks it's interesting.
3 replies
MathRook7817
Today at 2:19 AM
Jack_w
5 hours ago
J
H
Inequalities
lgx57   2
N 6 hours ago by sqing
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
2 replies
lgx57
Today at 7:43 AM
sqing
6 hours ago
J
H
solve the system of equations
Havu   1
N 6 hours ago by Mathzeus1024
Solve the system of equations:
\[\begin{cases} 3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\ 2x+\dfrac{1}{x-y}=4 \end{cases}\]
1 reply
Havu
Today at 7:52 AM
Mathzeus1024
6 hours ago
J
H
Inequalities
sqing   7
N Today at 8:12 AM by sqing
Let $a,b,c\ge \frac{1}{2}$ and $\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le 1. $ Prove that
$$a+b+c\geq 2$$Let $a,b,c\ge \frac{1}{2}$ and $ \left(a+\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(a+\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\le \frac{9}{2}. $ Prove that
$$a^2+b^2+c^2\geq 1$$Let $a,b\ge \frac{1}{2}$ and $ \left( \frac{1}{a}-\frac{1}{b}+2\right)\left( \frac{1}{b}-\frac{1}{a}+2\right) \le \frac{20}{9}. $ Prove that
$$ a+b\geq 2$$Let $a,b\ge \frac{1}{2}$ and $a^2+b^2=1. $ Prove that
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
7 replies
sqing
Mar 15, 2025
sqing
Today at 8:12 AM
J
H
Quadratic Equation Problems
Saucepan_man02   7
N Today at 5:53 AM by soryn
P1) Let $\alpha < \beta < \gamma$ be the roots of $ax^3+bx^2+cx+d=0$ with $7a+3b>0, 3a+2b<0$ and $\alpha(\beta+1)+\beta(\gamma+1)+\gamma(1+\alpha)=0$.
If [.] denotes the greatest integer function, then the number of possible values of $[|3 \alpha |] + [|6 \beta|] + [|9 \gamma |]$ is equal to
7 replies
Saucepan_man02
Mar 3, 2025
soryn
Today at 5:53 AM
J
H
Trash Trig Sum
P_Groudon   4
N Today at 5:41 AM by invisibleman
Find the smallest positive integer $n$ such that $$\sum_{k=0}^{298}\sin(k^2 + 2k + 2)\sin(2k + 2) = \frac{\cos(1) - \cos(n)}{2},$$where degrees are used.
4 replies
P_Groudon
Monday at 5:34 PM
invisibleman
Today at 5:41 AM
J
H
Chessboard
Ecrin_eren   2
N Today at 5:24 AM by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

2 replies
Ecrin_eren
Yesterday at 8:55 PM
Ecrin_eren
Today at 5:24 AM
J
H
Inequality
BaCaPhe   1
N Today at 4:43 AM by lbh_qys
$\begin{array}{l} a,b,c \ge 1,ab + bc + ca \ge 4 + abc\\ {a^2} + {b^2} + {c^2} \ge 9? \end{array}$
I tried:
$\begin{array}{l} * {\text{In three numbers }}a,b,c \ge 1,{\text{ at least 2 numbers }} \ge {\text{2 or}} \le {\text{2}}\\ {\text{WLOG assume they are }}a{\text{ and }}b \Rightarrow \left( {a - 2} \right)\left( {b - 2} \right) \ge 0 \Leftrightarrow 4c + abc \ge 2ac + 2bc\\ \Leftrightarrow 2ac + 2bc \le 4c - 4 + 4 + abc \le 4c - 4 + ab + bc + ca \Leftrightarrow ac + bc \le 4c - 4 + ab \end{array}$
How do I process further?
1 reply
BaCaPhe
Today at 3:13 AM
lbh_qys
Today at 4:43 AM
J
H
Inequalities
sqing   21
N Today at 4:36 AM by sqing
Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=1. $ Prove that
$$(a^2-a+1)(b^2-b+1) \geq 9$$$$ (a^2-a+b+1)(b^2-b+a+1) \geq 25$$Let $ a,b>0 $ and $ \frac{1}{a}+\frac{1}{b}=\frac{2}{3}. $ Prove that
$$(a+8)(a^2-a+b+2)(b^2-b+5)\geq1331$$$$(a+10)(a^2-a+b+4)(b^2-b+7)\geq2197$$
21 replies
sqing
Mar 10, 2025
sqing
Today at 4:36 AM
J
H
Inequalities
sqing   6
N Today at 4:16 AM by sqing
Let $ a,b $ be real numbers such that $ a + b \geq |ab + 1|. $ Prove that$$ a^3 + b^3 \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 2(a + b ) \geq |ab + 1|. $ Prove that$$26( a^3 + b^3) \geq |a^3 b^3 + 1|$$Let $ a,b $ be real numbers such that $ 4(a + b) \geq 3|ab + 1|. $ Prove that$$148(a^3 + b^3) \geq27 |a^3 b^3 + 1|$$
6 replies
sqing
Mar 8, 2025
sqing
Today at 4:16 AM
J
H
J
H
Algebra-2
JetFire008   1
N Mar 17, 2025 by joeym2011
Prove that if $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$, then $f(x^n)$ is divisible by $x^n-1$.
1 reply
JetFire008
Mar 16, 2025
joeym2011
Mar 17, 2025
J
Algebra-2
G H J
Reveal topic tags
Polynomials
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.
JetFire008
95 posts
JetFire008
#1 h  Mar 16, 2025, 3:23 PM
Y by
Prove that if $f(x)$ is a polynomial such that $f(x^n)$ is divisible by $x-1$, then $f(x^n)$ is divisible by $x^n-1$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
joeym2011
453 posts
joeym2011
#2 h  Mar 17, 2025, 2:11 AM
Y by
For any $n$th root of unity $\omega$, we have $f\left(\omega^n\right)=f(1)=f\left(1^n\right)=0$, so $f\left(x^n\right)$ has all such $\omega$ as roots.
Z K Y
N Quick Reply
G
H
=
a