Live Discussion!
Game Jam: An Ultrametric Number Guessing Game is going on now!

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
IMO Shortlist 2011, G4
WakeUp   135
N 39 minutes ago by LHE96
Source: IMO Shortlist 2011, G4
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.

Proposed by Ismail Isaev and Mikhail Isaev, Russia
135 replies
WakeUp
Jul 13, 2012
LHE96
39 minutes ago
Find all natural numbers n
Iwanttostudymathbetter   2
N an hour ago by Feita
Find all natural numbers $n$ that satisfy:
$\sigma(n)-\varphi(n)=n+4$
2 replies
Iwanttostudymathbetter
Mar 9, 2025
Feita
an hour ago
n+1 subsets
sturdyoak2012   3
N an hour ago by ostriches88
Suppose you have $n+1$ subsets of $\{1, 2, \ldots, n\}$ such that any two subsets have an intersection size of exactly one. Show that two of these subsets must be the same.
3 replies
sturdyoak2012
Sep 20, 2020
ostriches88
an hour ago
Third degree and three variable system of equations
MellowMelon   59
N an hour ago by lpieleanu
Source: USA TST 2009 #7
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]

Razvan Gelca.
59 replies
MellowMelon
Jul 18, 2009
lpieleanu
an hour ago
Automorphic Characteristic
KHOMNYO2   3
N Today at 4:51 PM by GreenKeeper
Given a group G where $|G| = p + 1$ for some odd prime $p$. It is known that $p \mid |Aut(G)|$. Prove that $p$ must be in the form of $4k + 3$, where $k$ is an integer. Give a group $|G|$ as an example that satisfies the property.
3 replies
KHOMNYO2
Feb 5, 2025
GreenKeeper
Today at 4:51 PM
expected value question
straight   2
N Today at 4:12 PM by grupyorum
Given positive reals $x_1< x_2, \dots< x_n$ randomly and independently picked in $[0,1]$ (you order them after picking). If you define $x_0 = 0$ and $x_{n+1} = 1$, what is
\[\mathbb{E}(\sum_{i=0}^{n}(x_{i+1} - x_i)^2))\]
2 replies
straight
Sep 28, 2024
grupyorum
Today at 4:12 PM
probability
nguyenalex   6
N Today at 4:07 PM by grupyorum
Let $(W_n )_{n\geq 1}$ be a sequence of independent random variables with standard normal distribution. Estimate the probability that $W_1^2 +W_2^2 +...+ W_{n} ^ 2 < n + 2\sqrt{n}$ .
6 replies
nguyenalex
Oct 26, 2024
grupyorum
Today at 4:07 PM
probability
ILOVEMYFAMILY   2
N Today at 3:39 PM by grupyorum
Let $(X_k)$ be a sequence of independent random variables with distribution $X_k \sim \text{Exp}(k)$. Define $M_n = \min_{1 \leq k \leq n} X_k$. Prove that $$\frac{n(n+1)M_n}{\ln n} \overset{P}{\rightarrow} 0$$and $$nM_n \overset{\text{a.s}}{\rightarrow} 0$$as $n \to \infty$.
2 replies
ILOVEMYFAMILY
Oct 29, 2024
grupyorum
Today at 3:39 PM
Convergence in distribution
Ernest532   4
N Today at 3:10 PM by grupyorum
Let $\{X_i\}$ be i.i.d with pdf $\frac{1}{\lvert x\rvert^3}\mathbb{I}_{\{\lvert x\rvert>1\}}$. Prove that $$\frac{S_n}{\sqrt{n\log(n)}}\xrightarrow[n\to\infty]{\text{d}}\mathcal{N}(0,1).$$
4 replies
Ernest532
Jun 30, 2025
grupyorum
Today at 3:10 PM
Putnam 2017 A2
Kent Merryfield   29
N Today at 11:42 AM by Assassino9931
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
29 replies
Kent Merryfield
Dec 3, 2017
Assassino9931
Today at 11:42 AM
Analysis
ratatuy   7
N Today at 9:28 AM by Mathzeus1024
Source: Own
$i)$Prove that
$\sum_{k=0}^{+\infty}{\frac{(-1)^k}{2k+1}}=\frac{\pi}{4}$
$ii)$Find
$\sum_{k=0}^{+\infty}{\frac{(-1)^k}{2k+2}}=?$
$iii)$Find
$\sum_{n=1}^{+\infty}{\frac{(-1)^n}{n}\left(\frac{1}{n+1}-\frac{1}{n+3}+\frac{1}{n+5}+...+\frac{(-1)^k}{n+1+2k}+...\right)}=?$
7 replies
ratatuy
Jul 3, 2020
Mathzeus1024
Today at 9:28 AM
Integer part
P0tat0b0y   1
N Today at 8:06 AM by P0tat0b0y
Source: own
Calculate the $\left[ {{\left( \prod\limits_{k=1}^{n}{\frac{2k}{2k-1}} \right)}^{2}}-\pi n \right]$, for positive $n$ inteher, where $[a]$ it is the integer part of the number $a$!
1 reply
P0tat0b0y
Jul 27, 2025
P0tat0b0y
Today at 8:06 AM
Differentiate the following function with respect to $x:$ $arccos\bigg(\frac{1-x
Vulch   1
N Today at 8:04 AM by alexheinis
Differentiate the following function with respect to $x:$
$\text{arccos}\bigg(\frac{1-x^2}{1+x^2}\bigg),~0<x<1.$

My attempts are in the following attachment.I got the answer $\frac{2x^2}{1+x^2},$ but the answer in book is given as $\frac{2}{1+x^2}.$ Where is my faults?
1 reply
Vulch
Today at 4:33 AM
alexheinis
Today at 8:04 AM
Hard sequence problem
P0tat0b0y   1
N Today at 8:04 AM by P0tat0b0y
Please help! This is not homework! But it would be very important to get an answer!
Is there a sequence ${{a}_{n}}$ for which the following 4 conditions are met:
(1) ${{a}_{1}}\le 4-\pi $
(2) $3,(5)-\frac{{{a}_{2}}}{2}\ge \pi$
(3) $\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{n}=0$
(4) ${{(2n+2)}^{2}}{{a}_{n}}\ge {{(2n+1)}^{2}}{{a}_{n+1}}+(n+1)\pi ,\forall n\ge {{n}_{0}}$
Thanks in advance!
1 reply
P0tat0b0y
Yesterday at 9:08 AM
P0tat0b0y
Today at 8:04 AM
Combi Algorithm/PHP/..
CatalanThinker   1
N May 28, 2025 by CatalanThinker
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
1 reply
CatalanThinker
May 28, 2025
CatalanThinker
May 28, 2025
Combi Algorithm/PHP/..
G H J
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CatalanThinker
13 posts
#1
Y by
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
CatalanThinker
13 posts
#2
Y by
Any ideas?
Z K Y
N Quick Reply
G
H
=
a