Plan ahead for the next school year. Schedule your class today!

G
Topic
First Poster
Last Poster
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
Divsibility (combinatorics)
cmtappu96   8
N 16 minutes ago by MODBreathing_FirstForm
Find the number of $4$-digit numbers (in base $10$) having non-zero digits and which are divisible by $4$ but not by $8$.
8 replies
cmtappu96
Dec 5, 2010
MODBreathing_FirstForm
16 minutes ago
combinatorial clusters
Cats_on_a_computer   0
23 minutes ago
Source: THM, Richard Earl
Let $m,n$ be positive integers. For each $k=1,2,\dots,m$, define
\[
\Delta_k \;=\;
\Bigl\lfloor \tfrac{k\,n}{m}\Bigr\rfloor
\;-\;
\Bigl\lfloor \tfrac{(k-1)\,n}{m}\Bigr\rfloor.
\]Observe that $\Delta_k\in\{0,1\}$ and $\sum_{k=1}^m\Delta_k=n$. We view the indices $1,2,\dots,m$ cyclically, so that $\Delta_{m+1}=\Delta_1$. A *cluster* is a maximal cyclic block of consecutive indices all of whose $\Delta_k=1$. If there are $d$ such clusters, write their lengths (in cyclic order) as
\[
\ell_1,\ell_2,\dots,\ell_d.
\]Prove the following:


(i) $d = \gcd(m,n)$.
(ii) Each cluster‐length $\ell_i$ equals either $\lfloor n/d\rfloor$ or $\lceil n/d\rceil$, and exactly
\[
    n \;-\; d\,\bigl\lfloor\tfrac{n}{d}\bigr\rfloor
  \]of the $\ell_i$s are equal to $\lceil n/d\rceil$ (the remaining $d - \bigl(n - d\lfloor n/d\rfloor\bigr)$ clusters have length $\lfloor n/d\rfloor$).
0 replies
Cats_on_a_computer
23 minutes ago
0 replies
Geometry finale: radical axis bisects D-altitude
v_Enhance   52
N 39 minutes ago by ravengsd
Source: USA TSTST 2016 Problem 6, by Danielle Wang
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$. Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$. Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.

Proposed by Danielle Wang
52 replies
v_Enhance
Jun 29, 2016
ravengsd
39 minutes ago
Know me where i m wrong
Not__Infinity   7
N an hour ago by wangyanliluke
So, the problem is


Show that there is no integer solution to this expression
(a+b)((a^2)+(b^2)) = 2001.

My solution:-

Now, factoring 2001 gives 3×23×29= 2001

Case 1: (a+b)((a^2)+(b^2)) = 3 ×667.

Solving for b gives and irrational. Let me give the proof.

Take a+b=3.....(¡) and ((a^2)+(b^2)) = 667...........(¡¡)

Rearrange (¡) for a, which is a = 3 - b.

substitute value of a in (¡¡) and you'll get an quadratic equation. Solving it will give and irrational value of b.

Doing this repeatedly, making cases from these factors above mentioned and comparing, we get that no integer value will be obtained. Hence proved.

Correct me if i am wrong.

Source is some thread which idk. But i find this interesting thing.
7 replies
Not__Infinity
Jul 12, 2025
wangyanliluke
an hour ago
NT By Probabilistic Method
EthanWYX2009   0
4 hours ago
Source: 2024 March 谜之竞赛-6
Given a positive integer \( k \) and a positive real number \( \varepsilon \), prove that there exist infinitely many positive integers \( n \) for which we can find pairwise coprime integers \( n_1, n_2, \cdots, n_k \) less than \( n \) satisfying
\[\text{gcd}(\varphi(n_1), \varphi(n_2), \cdots, \varphi(n_k)) \geq n^{1-\varepsilon}.\]Proposed by Cheng Jiang from Tsinghua University
0 replies
EthanWYX2009
4 hours ago
0 replies
a sequence of a polynomial
truongphatt2668   3
N Today at 3:28 AM by truongphatt2668
Let a sequence of polynomial defined by: $P_0(x) = x$ and $P_{n+1}(x) = -2xP_n(x) + P'_n(x), \forall n \in \mathbb{N}$.
Find: $P_{2017}(0)$
3 replies
truongphatt2668
Yesterday at 2:22 PM
truongphatt2668
Today at 3:28 AM
Minimum value
Martin.s   5
N Today at 2:52 AM by aaravdodhia
What is the minimum value of
$$
\frac{|a + b + c + d| \left( |a - b| |b - c| |c - d| + |b - a| |c - a| |d - a| \right)}{|a - b| |b - c| |c - d| |d - a|}
$$over all triples $a, b, c, d$ of distinct real numbers such that
$a^2 + b^2 + c^2 + d^2 = 3(ab + bc + cd + da).$

5 replies
Martin.s
Oct 17, 2024
aaravdodhia
Today at 2:52 AM
Aproximate ln(2) using perfect numbers
YLG_123   7
N Today at 12:04 AM by vincentwant
Source: Brazilian Mathematical Olympiad 2024, Level U, Problem 1
A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then:
\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]where the sums are taken over all prime divisors \(p\) of \(n\).
7 replies
YLG_123
Oct 12, 2024
vincentwant
Today at 12:04 AM
Putnam 2003 B3
btilm305   35
N Yesterday at 2:50 PM by SomeonecoolLovesMaths
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 $\le x$.)
35 replies
btilm305
Jun 23, 2011
SomeonecoolLovesMaths
Yesterday at 2:50 PM
Simple integuration
obihs   1
N Yesterday at 2:21 PM by Litvinov
Source: Own
Find the value of
$$\int_1^2\dfrac{\ln x}{(x^2-2x+2)^2}dx$$
1 reply
obihs
Yesterday at 8:44 AM
Litvinov
Yesterday at 2:21 PM
Estimating the Density
zqy648   0
Yesterday at 1:11 PM
Source: 2024 May 谜之竞赛-6
Given non-empty subset \( I \) of the set of positive integers, a positive integer \( n \) is called good if for every prime factor \( p \) of \( n \), \( \nu_p(n) \in I \). For a positive real number \( x \), let \( S(x) \) denote the number of good numbers not exceeding \( x \).

Determine all positive real numbers \( C \) and \( \alpha \) such that \(
\lim\limits_{x \to +\infty} \dfrac{S(x)}{x^\alpha} = C.
\)

Proposed by Zhenqian Peng, High School Affiliated to Renmin University of China
0 replies
zqy648
Yesterday at 1:11 PM
0 replies
Show that if \( d_3 < \frac{d_1}{3} \), then there exist two other positive diag
Martin.s   3
N Yesterday at 12:26 PM by Martin.s
Let
\[
D = \begin{bmatrix}
d_1 & 0 & 0 \\
0 & d_2 & 0 \\
0 & 0 & d_3
\end{bmatrix}, \quad
T = \begin{bmatrix}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 2
\end{bmatrix}
\]where \( d_1, d_2, d_3 \) are positive and \( d_1 \ge d_3 \).

Show that if \( d_3 < \frac{d_1}{3} \), then there exist two other positive diagonal matrices \( D_1 \) and \( D_2 \) such that \( D, D_1, D_2 \) are distinct but \( DT, D_1T, D_2T \) have the same eigenvalues.

Show also that if \( d_3 > \frac{d_1}{3} \) and \( D_1 \) is a positive diagonal matrix distinct from \( D \), then \( DT \) and \( D_1T \) must have different eigenvalues.
3 replies
Martin.s
Jun 23, 2025
Martin.s
Yesterday at 12:26 PM
Inequality
Martin.s   4
N Yesterday at 12:25 PM by Martin.s


For \( n = 2, 3, \dots \), the following inequalities hold:

\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{\sqrt{6}}{9}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \pi - \frac{\pi}{n},
\]
and

\[
-\frac{1}{3} \leq \frac{\sin(n\theta)}{n \sin \theta} \leq \frac{1}{5}
\quad \text{for } \frac{\pi}{n} \leq \theta \leq \frac{\pi}{2}.
\]
4 replies
Martin.s
Jun 23, 2025
Martin.s
Yesterday at 12:25 PM
Analytic Number Theory
zqy648   1
N Yesterday at 11:59 AM by zqy648
Source: 2024 December 谜之竞赛-6
For positive integer \( n \), define
\[S_n = \{(a, b) \in \mathbb{N}_+^2 \mid a, b < \sqrt{n} \text{ and } n \mid a^2 + b^3 + 1\}. \]Prove that there exists a positive real number \(\varepsilon\) such that for all integers \(n \geq 10\), \(
\left| S_n \right| < n^{\frac{1}{2} - \frac{\varepsilon}{\ln \ln n}}.
\)

Proposed by Yuxing Ye
1 reply
zqy648
Yesterday at 9:36 AM
zqy648
Yesterday at 11:59 AM
Difficult combinatorics problem
shactal   7
N May 19, 2025 by shactal
Can someone help me with this problem? Let $n\in \mathbb N^*$. We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$. We then denote $A_i>A_j$. A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?
7 replies
shactal
May 18, 2025
shactal
May 19, 2025
Difficult combinatorics problem
G H J
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.
shactal
9 posts
#1
Y by
Can someone help me with this problem? Let $n\in \mathbb N^*$. We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$. We then denote $A_i>A_j$. A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shactal
9 posts
#2
Y by
Someone?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aaravdodhia
2667 posts
#3
Y by
Isn't it $0$?

Note that $A_i$ beats $A_j$ when $E(d_i-d_j)>0$, where $E$ represents expected value and $d_i, d_j$ are the draws of players $i$ and $j$. That happens when $E(d_i) - E(d_j)>0$, or the sum of $A_i$'s collection is greater than the sum of $A_j$'s collection. In the problem, the sum of $A_1$'s collection must be greater than the sum of everybody else's, contradicting $A_n > A_1$.

This logic is due to the distribution being given prior to the players drawing and comparing their numbers. But if all distributions were considered at once, any pair $(i,j)$ would satisfy $A_i > A_j$ with equal probability $\tfrac12\left(\text{probability }E(d_i-d_j)\ne0\right)$, so the player's expected draws would all be the same. Hence there cannot be an order to $A_1\dots A_n$.

If this problem is from another source, I'd suggest reading their explanation. :)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shactal
9 posts
#4
Y by
Well, the thing is I don't have the solution and I would like to know the method to solve this type of problems
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ash_the_Bash07
1332 posts
#5
Y by
ok$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shactal
9 posts
#6
Y by
But I don't think the answer is $0$, because I already found some examples where the condition is satisfied
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shactal
9 posts
#7
Y by
Here is an example that satisfies the condition: Player $A$ has numbers $\{2,4,9\}$, player B has $\{1,6,8\}$ and player $C$ has $\{3,5,7\}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
shactal
9 posts
#8
Y by
If I can show that the events "$A$ wins against $B$" and "$B$ wins against $C$" are independent, then the problem is trivial. But how to prove this?
Z K Y
N Quick Reply
G
H
=
a