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

Plan ahead for the next school year. Schedule your class today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Wednesday, Jul 16 - Oct 29
Sunday, Aug 17 - Dec 14
Tuesday, Aug 26 - Dec 16
Friday, Sep 5 - Jan 16
Monday, Sep 8 - Jan 12
Tuesday, Sep 16 - Jan 20 (4:30 - 5:45 pm ET/1:30 - 2:45 pm PT)
Sunday, Sep 21 - Jan 25
Thursday, Sep 25 - Jan 29
Wednesday, Oct 22 - Feb 25
Tuesday, Nov 4 - Mar 10
Friday, Dec 12 - Apr 10

Prealgebra 2 Self-Paced

Prealgebra 2
Friday, Jul 25 - Nov 21
Sunday, Aug 17 - Dec 14
Tuesday, Sep 9 - Jan 13
Thursday, Sep 25 - Jan 29
Sunday, Oct 19 - Feb 22
Monday, Oct 27 - Mar 2
Wednesday, Nov 12 - Mar 18

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Tuesday, Jul 15 - Oct 28
Sunday, Aug 17 - Dec 14
Wednesday, Aug 27 - Dec 17
Friday, Sep 5 - Jan 16
Thursday, Sep 11 - Jan 15
Sunday, Sep 28 - Feb 1
Monday, Oct 6 - Feb 9
Tuesday, Oct 21 - Feb 24
Sunday, Nov 9 - Mar 15
Friday, Dec 5 - Apr 3

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Wednesday, Jul 2 - Sep 17
Sunday, Jul 27 - Oct 19
Monday, Aug 11 - Nov 3
Wednesday, Sep 3 - Nov 19
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Friday, Oct 3 - Jan 16
Sunday, Oct 19 - Jan 25
Tuesday, Nov 4 - Feb 10
Sunday, Dec 7 - Mar 8

Introduction to Number Theory
Tuesday, Jul 15 - Sep 30
Wednesday, Aug 13 - Oct 29
Friday, Sep 12 - Dec 12
Sunday, Oct 26 - Feb 1
Monday, Dec 1 - Mar 2

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Friday, Jul 18 - Nov 14
Thursday, Aug 7 - Nov 20
Monday, Aug 18 - Dec 15
Sunday, Sep 7 - Jan 11
Thursday, Sep 11 - Jan 15
Wednesday, Sep 24 - Jan 28
Sunday, Oct 26 - Mar 1
Tuesday, Nov 4 - Mar 10
Monday, Dec 1 - Mar 30

Introduction to Geometry
Monday, Jul 14 - Jan 19
Wednesday, Aug 13 - Feb 11
Tuesday, Aug 26 - Feb 24
Sunday, Sep 7 - Mar 8
Thursday, Sep 11 - Mar 12
Wednesday, Sep 24 - Mar 25
Sunday, Oct 26 - Apr 26
Monday, Nov 3 - May 4
Friday, Dec 5 - May 29

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)
Sat & Sun, Sep 13 - Sep 14 (1:00 - 4:00 PM PT/4:00 - 7:00 PM ET)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22
Friday, Aug 8 - Feb 20
Tuesday, Aug 26 - Feb 24
Sunday, Sep 28 - Mar 29
Wednesday, Oct 8 - Mar 8
Sunday, Nov 16 - May 17
Thursday, Dec 11 - Jun 4

Intermediate Counting & Probability
Sunday, Sep 28 - Feb 15
Tuesday, Nov 4 - Mar 24

Intermediate Number Theory
Wednesday, Sep 24 - Dec 17

Precalculus
Wednesday, Aug 6 - Jan 21
Tuesday, Sep 9 - Feb 24
Sunday, Sep 21 - Mar 8
Monday, Oct 20 - Apr 6
Sunday, Dec 14 - May 31

Advanced: Grades 9-12

Calculus
Sunday, Sep 7 - Mar 15
Wednesday, Sep 24 - Apr 1
Friday, Nov 14 - May 22

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Wednesday, Sep 3 - Nov 19
Tuesday, Sep 16 - Dec 9
Sunday, Sep 21 - Dec 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Oct 6 - Jan 12
Thursday, Oct 16 - Jan 22
Tues, Thurs & Sun, Dec 9 - Jan 18 (meets three times a week!)

MATHCOUNTS/AMC 8 Advanced
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 17 - Nov 9
Tuesday, Aug 26 - Nov 11
Thursday, Sep 4 - Nov 20
Friday, Sep 12 - Dec 12
Monday, Sep 15 - Dec 8
Sunday, Oct 5 - Jan 11
Tues, Thurs & Sun, Dec 2 - Jan 11 (meets three times a week!)
Mon, Wed & Fri, Dec 8 - Jan 16 (meets three times a week!)

AMC 10 Problem Series
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)
Sunday, Aug 10 - Nov 2
Thursday, Aug 14 - Oct 30
Tuesday, Aug 19 - Nov 4
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Mon, Wed & Fri, Oct 6 - Nov 3 (meets three times a week!)
Tue, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 10 Final Fives
Friday, Aug 15 - Sep 12
Sunday, Sep 7 - Sep 28
Tuesday, Sep 9 - Sep 30
Monday, Sep 22 - Oct 13
Sunday, Sep 28 - Oct 19 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, Oct 8 - Oct 29
Thursday, Oct 9 - Oct 30

AMC 12 Problem Series
Wednesday, Aug 6 - Oct 22
Sunday, Aug 10 - Nov 2
Monday, Aug 18 - Nov 10
Mon & Wed, Sep 15 - Oct 22 (meets twice a week!)
Tues, Thurs & Sun, Oct 7 - Nov 2 (meets three times a week!)

AMC 12 Final Fives
Thursday, Sep 4 - Sep 25
Sunday, Sep 28 - Oct 19
Tuesday, Oct 7 - Oct 28

AIME Problem Series A
Thursday, Oct 23 - Jan 29

AIME Problem Series B
Tuesday, Sep 2 - Nov 18

F=ma Problem Series
Tuesday, Sep 16 - Dec 9
Friday, Oct 17 - Jan 30

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, Aug 14 - Oct 30
Sunday, Sep 7 - Nov 23
Tuesday, Dec 2 - Mar 3

Intermediate Programming with Python
Friday, Oct 3 - Jan 16

USACO Bronze Problem Series
Wednesday, Sep 3 - Dec 3
Thursday, Oct 30 - Feb 5
Tuesday, Dec 2 - Mar 3

Physics

Introduction to Physics
Tuesday, Sep 2 - Nov 18
Sunday, Oct 5 - Jan 11
Wednesday, Dec 10 - Mar 11

Physics 1: Mechanics
Sunday, Sep 21 - Mar 22
Sunday, Oct 26 - Apr 26
0 replies
jwelsh
Jul 1, 2025
0 replies
J
H
Bonza functions
KevinYang2.71   58
N 33 minutes ago by matematica007
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[ f(a)~~\text{divides}~~b^a-f(b)^{f(a)} \]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
58 replies
KevinYang2.71
Jul 15, 2025
matematica007
33 minutes ago
J
H
too much tangencies these days...
kamatadu   2
N 35 minutes ago by mudkip42
Source: Cut The Knot
Let $\Omega$ be a circle and $\gamma_1,\gamma_2$ be circles internally tangent to $\Omega$ at $P$ and $Q$. Assume that $\gamma_1$ and $\gamma_2$ are also externally tangent at point $T$. Prove that the line through $P$ perpendicular to $PT$ meets line $QT$ on $\Omega$.
2 replies
kamatadu
Jan 20, 2023
mudkip42
35 minutes ago
J
H
The inekoalaty game
sarjinius   33
N an hour ago by maromex
Source: 2025 IMO P5
Alice and Bazza are playing the inekoalaty game, a two‑player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
[list]
[*] If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that
\[ x_1 + x_2 + \cdots + x_n \le \lambda n. \][*]If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
\[ x_1^2 + x_2^2 + \cdots + x_n^2 \le n. \][/list]
If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.

Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.

Proposed by Massimiliano Foschi and Leonardo Franchi, Italy
33 replies
sarjinius
Jul 16, 2025
maromex
an hour ago
J
H
Next term is sum of three largest proper divisors
vsamc   20
N an hour ago by ray66
Source: 2025 IMO P4
A proper divisor of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.

The infinite sequence $a_1, a_2, \cdots$ consists of positive integers, each of which has at least three proper divisors. For each $n\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.

Determine all possible values of $a_1$.

Proposed by Paulius Aleknavičius, Lithuania
20 replies
vsamc
Jul 16, 2025
ray66
an hour ago
J
H
Spectral radius
ILOVEMYFAMILY   1
N Today at 11:54 AM by alexheinis
Let $A \in \mathbb{R}^{n \times n}$. The spectral radius of $A$, denoted by $\rho(A)$, is defined as
\[ \rho(A) = \max_i |\lambda_i| \]where $\lambda_i$ are all the eigenvalues of the matrix $A$.
Let $A \in \mathbb{R}^{n \times n}$. There exists a norm $\|\cdot\|$ such that $\|A\| < 1$ if and only if the spectral radius of $A$ satisfies the condition $\rho(A) < 1$.
1 reply
ILOVEMYFAMILY
Yesterday at 1:41 PM
alexheinis
Today at 11:54 AM
J
H
Putnam 2014 A4
Kent Merryfield   37
N Today at 3:36 AM by numbertheory97
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
37 replies
Kent Merryfield
Dec 7, 2014
numbertheory97
Today at 3:36 AM
J
H
Find the value of $x$ in the following matrix problem: $\begin{bmatrix}x-5-1\en
Vulch   3
N Today at 3:06 AM by Etkan
Find the value of $x$ in the following matrix problem:

$\begin{bmatrix}x&-5&-1\end{bmatrix}\begin{bmatrix}1&0&2\\0&2&1\\2&0&3\end{bmatrix}\begin{bmatrix}x\\4\\1\end{bmatrix}=0$
3 replies
Vulch
Today at 1:58 AM
Etkan
Today at 3:06 AM
J
H
Putnam 2003 B3
btilm305   36
N Today at 2:36 AM by SirAppel
Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\](Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)
36 replies
btilm305
Jun 23, 2011
SirAppel
Today at 2:36 AM
J
H
irritative method
ILOVEMYFAMILY   3
N Today at 12:11 AM by ILOVEMYFAMILY
If $\|T\| < 1$ for any natural matrix norm and $c$ is a given vector, then the sequence $\{x^{(k)}\}_{k=0}^{\infty}$ defined by
\[ x^{(k)} = T x^{(k-1)} + c \]converges, for any $x^{(0)} \in \mathbb{R}^n$, to a vector $x \in \mathbb{R}^n$, with $x = Tx + c$, and the following error bounds hold:
a) $\|x - x^{(k)}\| \leq \|T\|^k \|x^{(0)} - x\|$
b) $\|x - x^{(k)}\| \leq \frac{\|T\|^k}{1 - \|T\|} \|x^{(1)} - x^{(0)}\|$
3 replies
ILOVEMYFAMILY
Yesterday at 9:43 AM
ILOVEMYFAMILY
Today at 12:11 AM
J
H
Eccentricity Sleuthing
Mathzeus1024   1
N Yesterday at 7:53 PM by vanstraelen
A nice problem involving conics.
1 reply
Mathzeus1024
Jul 15, 2025
vanstraelen
Yesterday at 7:53 PM
J
H
Numerical analysis
Tricky123   0
Yesterday at 4:09 PM
Q) the exponent n of binary digits gives a range of 0 to $2^{n}-1$
But how we get it if any one give me the concept of approach the problem? Help
0 replies
Tricky123
Yesterday at 4:09 PM
0 replies
J
H
Irritative methods
ILOVEMYFAMILY   0
Yesterday at 1:43 PM
Let $A \in \mathbb{R}^{n \times n}$. Prove that:
1) If $A$ is a diagonally dominant matrix, then both the Jacobi and Gauss-Seidel methods converge, and the Gauss-Seidel method converges faster in the sense that $\rho(T_{GS}) < \rho(T_J)$, where $T_{GS}$ and $T_J$ are the iteration matrices defined by $x^{(k+1)}=Tx^k+b$ for each method
2) If $A$ is a symmetric positive definite matrix, then both the Jacobi and Gauss-Seidel methods converge.
0 replies
ILOVEMYFAMILY
Yesterday at 1:43 PM
0 replies
J
H
analysis
We2592   1
N Yesterday at 12:56 AM by alexheinis
Q) find the value of the integration $I=\int_{a}^{b} \frac{e^{-|x|}}{1+(sinhx)^2}$
1 reply
We2592
Jul 17, 2025
alexheinis
Yesterday at 12:56 AM
J
H
Are all solutions normal ?
loup blanc   0
Jul 17, 2025
This post is linked to this one
https://artofproblemsolving.com/community/c7t290f7h3608120_matrix_equation
Let $Z=\{A\in M_n(\mathbb{C}) ; (AA^*)^2=A^4\}$.
If $A\in Z$ is a normal matrix, then $A$ is unitarily similar to $diag(H_p,S_{n-p})$,
where $H$ is hermitian and $S$ is skew-hermitian.
But are there other solutions? In other words, is $A$ necessarily normal?
I don't know the answer.
0 replies
loup blanc
Jul 17, 2025
0 replies
J
H
J
H
Algebra manipulation excercise
Marinchoo   3
N May 20, 2025 by compoly2010
Source: 2007 Bulgarian Autumn Math Competition, Problem 9.2
Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.
3 replies
Marinchoo
Mar 17, 2022
compoly2010
May 20, 2025
J
Algebra manipulation excercise
G H J
Reveal topic tags
algebra
Symmetric
L
G H BBookmark kLocked kLocked NReply
Source: 2007 Bulgarian Autumn Math Competition, Problem 9.2
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Marinchoo
407 posts
Marinchoo
#1 h  Mar 17, 2022, 5:32 PM
Y by
Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.
This post has been edited 1 time. Last edited by Marinchoo, Mar 17, 2022, 7:29 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
AlexCenteno2007
166 posts
AlexCenteno2007
#2 h  May 20, 2025, 2:17 AM
Y by
Solution:
By elementary symmetric polynomials we have S1=$a+b+c=0$
S2=-2$\sigma(2)$
We know that $\sigma(2)$ = ab+bc+ac
By definition we have:



\[ S(n) = \sigma_1 S_{n-1} - \sigma_2 S_{n-2} + \sigma_3 S_{n-3} \]
Replacing :
\
\[ S_4 = - \sigma_2 \cdot (-2\sigma_2) \]

Therefore, :

\[ \sigma_2 = \pm 5 \]
This post has been edited 6 times. Last edited by AlexCenteno2007, May 20, 2025, 2:24 AM
Reason: Technical failures
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
43230 posts
sqing
#3 h  May 20, 2025, 2:53 AM • 1 Y
Y by AlexCenteno2007
Let $a$, $b$, $c$ be real numbers such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Prove that $$ab+bc+ca=-5$$Let $a$, $b$, $c$ be real numbers such that $a+b+c=0$ and $a^4+b^4+c^4=2$. Prove that $$ab+bc+ca=-1$$Maybe...
This post has been edited 3 times. Last edited by sqing, May 20, 2025, 2:59 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
compoly2010
15 posts
compoly2010
#4 h  May 20, 2025, 3:45 AM
Y by
@2 above
Try and square the first condition you got and then subtract the case when it is positive 5. This makes that case impossible via sign arguments
Z K Y
N Quick Reply
G
H
=
a