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
Inequality
jokehim   1
N 4 minutes ago by no_room_for_error
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\sqrt{a^2+2abc+b^2}+\sqrt{b^2+2abc+c^2}+\sqrt{c^2+2abc+a^2}\le 6.$$Proposed by Phan Ngoc Chau
1 reply
jokehim
Today at 7:34 AM
no_room_for_error
4 minutes ago
J
H
Properties of a line
Cathong0324   2
N 21 minutes ago by XAN4
If we connect a triangle's incenter and circumcenter, what are the properties of this line?
2 replies
Cathong0324
Mar 5, 2024
XAN4
21 minutes ago
J
H
rhombus, line from bisector x circumcircle points (Puerto Rico TST 2024.7)
Equinox8   1
N 34 minutes ago by pooh123
Let $I$ be the incenter of $\triangle{ABC}$. Angle bisectors $BI$ and $CI$ intersect the circumcircle of $\triangle{ABC}$ at points $S$ and $T$, respectively ($S \neq B$ and $T \neq C$). Line $ST$ intersects sides $AB$ and $AC$ at points $K$ and $L$, respectively. Prove that quadrilateral $AKIL$ is a rhombus.

Note: A rhombus is a parallelogram with all sides equal.
1 reply
Equinox8
Mar 12, 2025
pooh123
34 minutes ago
J
H
Collinear proof
Wiselady   2
N an hour ago by Wiselady
Let ABC be any triangle. Let side BC pass through point C to point D such that CD=AC. Let P be the second intersection point of the circumscribed circle ACD with the circle with BC as the diameter. Let BP and AC meet at point E and let CP and AB meet at point F. Prove that D, E, F lie on the same straight line.
IMAGE
2 replies
Wiselady
Today at 1:33 AM
Wiselady
an hour ago
J
H
Happy 3.14
JetFire008   2
N an hour ago by whwlqkd
Happy PI day!! :clap: :trampoline:
2 replies
JetFire008
an hour ago
whwlqkd
an hour ago
J
H
Inequalities
sqing   0
an hour ago
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
$$\left(\frac{2}{a}+\frac{1}{b}-1\right)\left(\frac{2}{a}-\frac{1}{b}+1\right)\ge \frac{13}{3}$$
0 replies
sqing
an hour ago
0 replies
J
H
IOQM P2 2024
SomeonecoolLovesMaths   10
N an hour ago by BackToSchool
The number of four-digit odd numbers having digits $1,2,3,4$, each occuring exactly once, is:
10 replies
SomeonecoolLovesMaths
Sep 8, 2024
BackToSchool
an hour ago
J
H
Find a line in inequality gemetry
William_Mai   1
N 3 hours ago by William_Mai
M point is given inside a angle xOy. Draw a line d that passes through M and intersects Ox and Oy at the A và B with the smallest possible of AB.
(Note: line d not passing through O)

IMAGE
1 reply
William_Mai
Mar 9, 2025
William_Mai
3 hours ago
J
H
2017 BCSMC Round 2 #1 (BC)^2 - (AB)^2 = 1081 in right ABC
parmenides51   2
N 4 hours ago by BackToSchool
In right triangle $ABC$, $\angle ABC = 90^o$, $AC = 37$, and $(BC)^2 - (AB)^2 = 1081$. Find $AB + BC$.
2 replies
parmenides51
Jan 24, 2024
BackToSchool
4 hours ago
J
H
Inequalities
sqing   10
N 4 hours ago by sqing
Let $ a,b $ be reals such that $ a^2+b^2=1. $ Prove that
$$\sqrt{ a+1}+ \sqrt{ b+1}+k \sqrt{ 2ab+1} \leq \sqrt{2}(k+ \sqrt{ 2+\sqrt{2}})$$Where $ k>0. $
10 replies
sqing
Mar 9, 2025
sqing
4 hours ago
J
H
Inequalities from SXTX
sqing   6
N 6 hours ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
6 replies
sqing
Feb 18, 2025
sqing
6 hours ago
J
H
Difficult Geometry Optimization Problem.
ReticulatedPython   4
N Today at 8:00 AM by bhontu
Consider three concentric circles with radii $x$, $y$, and $z.$ Point $X$ is chosen on the circle with radius $x$, point $Y$ is chosen on the circle with radius $y$, and point $Z$ is chosen on the circle with radius $z.$ Given that $x<y<z$, find the maximum and minimum values of $$XY^2+YZ^2+XZ^2$$in terms of $x$, $y$, and $z.$
4 replies
ReticulatedPython
Yesterday at 3:09 PM
bhontu
Today at 8:00 AM
J
H
2017 BCSMC Round 2 #3 yellow pig walks on a number line
parmenides51   3
N Today at 4:20 AM by MathPerson12321
A yellow pig walks on a number line starting at $17$. Each step the pig has probability $\frac{8}{17}$ of moving $1$ unit in the positive direction and probability $\frac{9}{17}$ of moving $1$ unit in the negative direction. Find the expected number of steps until the yellow pig visits $0$.
3 replies
parmenides51
Jan 24, 2024
MathPerson12321
Today at 4:20 AM
J
H
number theory
IOQMaspirant   6
N Today at 3:19 AM by Yiyj1
Prove 6|(a + b + c) if and only if 6|(a^3 + b^3 + c^3)
6 replies
IOQMaspirant
Mar 11, 2025
Yiyj1
Today at 3:19 AM
J
2017 BCSMC Round 2 #1 (BC)^2 - (AB)^2 = 1081 in right ABC
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.
parmenides51
30627 posts
parmenides51
#1 h  Jan 24, 2024, 4:45 PM
Y by
In right triangle $ABC$, $\angle ABC = 90^o$, $AC = 37$, and $(BC)^2 - (AB)^2 = 1081$. Find $AB + BC$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CubeAlgo15
35 posts
CubeAlgo15
#2 h  Yesterday at 10:18 PM
Y by
Add \( 2AB \) to both sides, then we have \( BC^2 + AB^2 = 1081 + 2(AB)^2 \) rearranging gives \( 37^2 - 1081 = 2(AB)^2 \)
So AB = 12 and BC = 35, thus our answer is \( \boxed{47} \).
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BackToSchool
1638 posts
BackToSchool
#3 h  4 hours ago
Y by
$$(BC)^2 - (AB)^2 = 1081$$$$(BC)^2 + (AB)^2 = (AC)^2 = 37^2 = 1369$$$$(BC)^2 = \frac {1081+1369}{2} = 1225, BC = 35$$$$(AB)^2 = \frac {- 1081+1369}{2} = 144, AB = 12$$$$AB + BC = \boxed {47}$$
This post has been edited 1 time. Last edited by BackToSchool, 2 hours ago
Z K Y
N Quick Reply
G
H
=
a