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
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
AD+BC=AB in a cyclic quadrilateral
orl   39
N 7 minutes ago by Kempu33334
Source: 1985, Day 1, Problem 1
A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.
39 replies
orl
Nov 11, 2005
Kempu33334
7 minutes ago
Functions
Math2030   0
40 minutes ago
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying
\[
f\left(x^2 + xy^2 + y^2\right) = 2x^2 f(y) + 2x f(f(y)) + f(-x^2 - xy^2) + f(y^2)
\]for all \( x, y \in \mathbb{R} \).
0 replies
Math2030
40 minutes ago
0 replies
Collinearity with orthocenter
liberator   191
N an hour ago by Kempu33334
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
191 replies
+1 w
liberator
Jan 4, 2016
Kempu33334
an hour ago
sequence with sum of digits
silouan   5
N an hour ago by ringodanya
Source: All Russian MO 2015, grade 10, problem 4
We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$.
Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good?
A. Antropov
5 replies
silouan
Aug 8, 2015
ringodanya
an hour ago
Nice and Difficult Geometry (Collinearity)
RANDOM__USER   1
N an hour ago by mathnerd_101
Source: Own
Let \( D \) be an arbitrary point on the side \( BC \) of triangle \( \triangle ABC \). Let \( E \) and \( F \) be the intersections of the lines through \( D \), parallel to \( AC \) and \( AB \), with \( AB \) and \( AC \), respectively. Let \( G \) be the intersection point of the circumcircle of \( \triangle AFE \) with the circumcircle of \( \triangle ABC \). Let \( M \) be the midpoint of \( BC \), and let \( X \) be the intersection point of line \( AM \) with the circumcircle of \( \triangle ABC \). Prove that \( X \), \( D \), and \( G\) are collinear.

IMAGE
1 reply
RANDOM__USER
an hour ago
mathnerd_101
an hour ago
moving stones
outback   2
N an hour ago by Wasdshift
Source: Russia 1994
On the vertices of a convex $ n$-gon are put $ m$ stones, $ m > n$. In each move we can choose two stones standing at the same vertex and move them to
the two distinct adjacent vertices. After $ N$ moves the number of stones at each vertex was the same as at the beginning. Prove that $ N$ is divisible by $ n$.
2 replies
outback
Oct 6, 2008
Wasdshift
an hour ago
N.T with polynomial
X.Allaberdiyev   4
N an hour ago by jonh_malkovich
Source: IMSC P5
Let $Q(n)$ be a polynomial with integer coefficents. We call the positive integer $n$ $special$ if for every prime $p$: \[
p \mid \sigma(n) + 1 \quad \text{if and only if} \quad p \mid Q(n)
\]Then prove that there are infinitely many positive integers $n$ such that neither $n$ nor $n+1$ is special.
4 replies
X.Allaberdiyev
Jul 7, 2025
jonh_malkovich
an hour ago
2025 IMO TEAMS
Oksutok   56
N 2 hours ago by Shayan-TayefehIR
Good Luck in Sunshine Coast, Australia
56 replies
Oksutok
May 14, 2025
Shayan-TayefehIR
2 hours ago
Iran National Math Olympiad-Day 1-Problem 2
Amir Hossein   13
N 2 hours ago by de-Kirschbaum
There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.
13 replies
Amir Hossein
Apr 30, 2010
de-Kirschbaum
2 hours ago
Is this sum an integer?
AwesomeYRY   15
N 2 hours ago by numbersandnumbers
Source: TSTST 2021/9
Let $q=p^r$ for a prime number $p$ and positive integer $r$. Let $\zeta = e^{\frac{2\pi i}{q}}$. Find the least positive integer $n$ such that
\[\sum_{\substack{1\leq k\leq q\\ \gcd(k,p)=1}} \frac{1}{(1-\zeta^k)^n}\]is not an integer. (The sum is over all $1\leq k\leq q$ with $p$ not dividing $k$.)

Victor Wang
15 replies
1 viewing
AwesomeYRY
Jan 17, 2022
numbersandnumbers
2 hours ago
Easy but nice NT with lcm
a_507_bc   8
N 2 hours ago by quantam13
Source: ARO Regional stage 2023 9.6=10.6=11.6
Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?
8 replies
a_507_bc
Feb 16, 2023
quantam13
2 hours ago
Orhocenter of cevian triangle of P wrt orthic triangle, K742
kosmonauten3114   0
2 hours ago
Source: My own
Let $\triangle{ABC}$ be a scalene triangle with orthic triangle $\triangle{H_AH_BH_C}$.
Let $P$ be a point on the orthic inconic of $\triangle{ABC}$.
Let $\triangle{P_AP_BP_C}$ be the cevian triangle of $P$ wrt $\triangle{H_AH_BH_C}$.
Let $H_P$ be the orthocenter of $\triangle{P_AP_BP_C}$.

Prove that $H_P$ lies on $\text{K742}$ of $\triangle{ABC}$. $\text{K742}$ is a nodal cubic with node $\text{X(155)}$, through 4 in/excenters, nine-point center, $H_A$, $H_B$, $H_C$, $\text{X(155)}$, $\text{X(185)}$.
0 replies
kosmonauten3114
2 hours ago
0 replies
Game Theory on Polygon Coloring
steven_zhang123   3
N 2 hours ago by Giabach298
Source: 2025 CTST P10
Given an odd integer $n \geq 3$. Let $V$ be the set of vertices of a regular $n$-gon, and $P$ be the set of all regular polygons formed by points in $V$. For instance, when $n=15$, $P$ consists of $1$ regular $15$-gon, $3$ regular pentagons, and $5$ regular triangles.

Initially, all points in $V$ are uncolored. Two players, $A$ and $B$, play a game where they take turns coloring an uncolored point, with player $A$ starting and coloring points red, and player $B$ coloring points blue. The game ends when all points are colored. A regular polygon in $P$ is called $\textit{good}$ if it has more red points than blue points.

Find the largest positive integer $k$ such that no matter how player $B$ plays, player $A$ can ensure that there are at least $k$ $\textit{good}$ polygons.
3 replies
2 viewing
steven_zhang123
Mar 10, 2025
Giabach298
2 hours ago
ISI UGB 2025 P4
SomeonecoolLovesMaths   16
N 2 hours ago by Safal
Source: ISI UGB 2025 P4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
16 replies
SomeonecoolLovesMaths
May 11, 2025
Safal
2 hours ago
Easy geo
kooooo   3
N May 17, 2025 by Blackbeam999
Source: own
In triangle $ABC$, let $O$ and $H$ be the circumcenter and orthocenter, respectively. Let $M$ and $N$ be the midpoints of $AC$ and $AB$, respectively, and let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to the opposite sides, respectively. Show that if $X$ is the intersection of $MN$ and $DE$, then $AX$ is perpendicular to $OH$.
3 replies
kooooo
Jul 31, 2024
Blackbeam999
May 17, 2025
Easy geo
G H J
G H BBookmark kLocked kLocked NReply
Source: own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
kooooo
21 posts
#1
Y by
In triangle $ABC$, let $O$ and $H$ be the circumcenter and orthocenter, respectively. Let $M$ and $N$ be the midpoints of $AC$ and $AB$, respectively, and let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to the opposite sides, respectively. Show that if $X$ is the intersection of $MN$ and $DE$, then $AX$ is perpendicular to $OH$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aidenkim119
41 posts
#2
Y by
Sx point line on nine point circle
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexmay
1 post
#3
Y by
Hint : (AMNO) (ADHE) (MNDE)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Blackbeam999
58 posts
#4
Y by
How can I prove this
Z K Y
N Quick Reply
G
H
=
a