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
Counting in two different ways
bamboozled   0
3 minutes ago
For a positive integer $n$ show that
$$\sum_{i=0}^{\lfloor \frac{n}{2} \rfloor} 2^{n-2i} \binom{n}{n-2i} \binom{2i}{i} = \binom{2n}{n}$$
0 replies
bamboozled
3 minutes ago
0 replies
J
H
Problem Regarding Measure Theory
Safal   2
N 31 minutes ago by Safal
Source: My Friend and Me
Let $(X,M,\nu)$ be a measure space, and $E\subseteq X$.
Let $$f:E\to[0,\infty]$$be a measurable function, and $f\in L^{1}(\nu)$. Suppose $$E_1\subset E_2\subset E_3\subset\cdots$$such that $\bigcup_{i=1}^{\infty} E_{n}=E$.
Is it true that $$\lim_{n\rightarrow\infty} \int_{E_n} f d\nu =\int_{E} f d\nu?$$
I think
2 replies
Safal
an hour ago
Safal
31 minutes ago
J
H
Problem on distinct prime divisors of P(1),...,P(n)
IAmTheHazard   0
4 hours ago
Find all nonnegative real numbers $\lambda$ such that there exists an integer polynomial $P$ with no integer roots and a constant $c>0$ such that
$$\prod_{i=1}^n P(i)=P(1)\cdot P(2)\cdots P(n)$$has at least $cn^{\lambda}$ distinct prime divisors for all positive integers $n$.
0 replies
IAmTheHazard
4 hours ago
0 replies
J
H
How To Solve This Problem Using Multivariable Calculus
temp510875   4
N 4 hours ago by rchokler
If $|x+y|\hspace{0.1cm}+\hspace{0.1cm}|x-y|\hspace{0.1cm}=\hspace{0.1cm}2$, find the minimum value of the the expression $2x^2-xy+2y^2$.
4 replies
temp510875
Oct 21, 2019
rchokler
4 hours ago
J
H
Geo Mock #10
Bluesoul   2
N 6 hours ago by Bluesoul
Consider acute $\triangle{ABC}$ with $AB=10$, $AC<BC$ and area $135$. The circle $\omega$ with diameter $AB$ meets $BC$ at $E$. Let the orthocenter of the triangle be $H$, connect $CH$ and extend to meet $\omega$ at $N$ such that $NC>HC$ and $NE$ is the diameter of $\omega$. Draw the circumcircle $\Gamma$ of $\triangle{AHB}$, chord $XY$ of $\Gamma$ is tangent to $\omega$ and it passes through $N$, compute $XY$.
2 replies
Bluesoul
Apr 1, 2025
Bluesoul
6 hours ago
J
H
Inequalities
sqing   4
N Yesterday at 3:28 PM by sqing
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
4 replies
sqing
Apr 1, 2025
sqing
Yesterday at 3:28 PM
J
H
Geo Mock #9
Bluesoul   1
N Yesterday at 3:19 PM by vanstraelen
Consider $\triangle{ABC}$ with $AB=12, AC=22$. The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$. Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$. Extend $AT$ to $BC$ at $R$, given $AR=16$, find $[ABC]$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 3:19 PM
J
H
Regarding IMO prepartion
omega2007   0
Yesterday at 3:14 PM
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
0 replies
omega2007
Yesterday at 3:14 PM
0 replies
J
H
Geo Mock #6
Bluesoul   1
N Yesterday at 1:59 PM by vanstraelen
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 1:59 PM
J
H
Congruence
Ecrin_eren   1
N Yesterday at 1:39 PM by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

1 reply
Ecrin_eren
Thursday at 10:34 AM
Ecrin_eren
Yesterday at 1:39 PM
J
H
Probability
Ecrin_eren   1
N Yesterday at 1:38 PM by Ecrin_eren
In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

1 reply
Ecrin_eren
Thursday at 11:21 AM
Ecrin_eren
Yesterday at 1:38 PM
J
H
Excalibur Identity
jjsunpu   9
N Yesterday at 12:21 PM by fruitmonster97
proof is below
9 replies
jjsunpu
Thursday at 3:27 PM
fruitmonster97
Yesterday at 12:21 PM
J
H
.problem.
Cobedangiu   2
N Yesterday at 12:06 PM by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
2 replies
Cobedangiu
Yesterday at 6:20 AM
Lankou
Yesterday at 12:06 PM
J
H
Inequalities
sqing   23
N Yesterday at 11:43 AM by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
23 replies
sqing
Mar 26, 2025
sqing
Yesterday at 11:43 AM
J
H
J
H
complex roots of polynomial and inequality for coefficients
su7e   5
N May 17, 2021 by XbenX
polynomial $P(z)=a_nz^n+\dots+ a_1z+a_0\in\mathbb{R}[x]$, $\deg P=n\ge 3$, has all complex roots in a set $\{z\in\mathbb{C}: \mbox{Re}\,z<0\}$.
show that $a_ka_{k+3}<a_{k+1}a_{k+2}$ for $k=0,1,\dots, n-3$.
5 replies
su7e
Nov 29, 2012
XbenX
May 17, 2021
J
complex roots of polynomial and inequality for coefficients
G H J
Reveal topic tags
algebra
polynomial
inequalities
complex analysis
complex analysis unsolved
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.
su7e
16 posts
su7e
#1 h  Nov 29, 2012, 11:42 PM • 2 Y
Y by Adventure10, Mango247
polynomial $P(z)=a_nz^n+\dots+ a_1z+a_0\in\mathbb{R}[x]$, $\deg P=n\ge 3$, has all complex roots in a set $\{z\in\mathbb{C}: \mbox{Re}\,z<0\}$.
show that $a_ka_{k+3}<a_{k+1}a_{k+2}$ for $k=0,1,\dots, n-3$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CJA
155 posts
CJA
#2 h  Nov 13, 2019, 1:32 AM • 2 Y
Y by Adventure10, Mango247
any ideas?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GreenKeeper
1684 posts
GreenKeeper
#3 h  Nov 13, 2019, 6:41 AM • 1 Y
Y by Adventure10
It's probably related to Routh-Hurwitz theorem.

https://en.wikipedia.org/wiki/Hurwitz_matrix
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CJA
155 posts
CJA
#4 h  Nov 13, 2019, 11:52 AM • 1 Y
Y by Adventure10
GreenKeeper wrote:
It's probably related to Routh-Hurwitz theorem.

https://en.wikipedia.org/wiki/Hurwitz_matrix

thank you,this problem is really related to Routh-Hurwitz theorem,it is about Hurwitz polynomial,and I know the problem is called xie xvkai stability criterion（proposed in 1963） now.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Xiaolin
36 posts
Xiaolin
#6 h  May 17, 2021, 6:44 AM
Y by
Any solutions?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
XbenX
590 posts
XbenX
#7 h  May 17, 2021, 7:20 PM • 1 Y
Y by GreenKeeper
It is IMC 2003 P6; you can find a solution on the official page.
Z K Y
N Quick Reply
G
H
=
a