Stay ahead of learning milestones! Enroll in a class over the summer!

Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3 M G
BBookmark  VNew Topic kLocked
Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
3 M G
BBookmark  VNew Topic kLocked
G
Topic
First Poster
Last Poster
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
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
9 USA(J)MO Grading Poll
elasticwealth   5
N a minute ago by wuwang2002
Please vote honestly. If you did not compete in the USA(J)MO, please do not vote.
5 replies
elasticwealth
2 hours ago
wuwang2002
a minute ago
RIP BS2012
gavinhaominwang   10
N 8 minutes ago by Ilikeminecraft
Rip BS2012, I hope you come back next year stronger and prove everyone wrong.
10 replies
gavinhaominwang
5 hours ago
Ilikeminecraft
8 minutes ago
2025 USA(J)MO Cutoff Predictions
KevinChen_Yay   109
N 22 minutes ago by Alpaca31415
What do y'all think JMO winner and MOP cuts will be?

(Also, to satisfy the USAMO takers; what about the bronze, silver, gold, green mop, blue mop, black mop?)
109 replies
KevinChen_Yay
Mar 21, 2025
Alpaca31415
22 minutes ago
2025 ELMOCOUNTS - Mock MATHCOUNTS Nationals
vincentwant   106
N an hour ago by mathkiddus
text totally not copied over from wmc (thanks jason <3)
Quick Links:
[list=disc]
[*] National: (Sprint) (Target) (Team) (Sprint + Target Submission) (Team Submission) [/*]
[*] Miscellaneous: (Leaderboard) (Sprint + Target Private Discussion Forum) (Team Discussion Forum)[/*]
[/list]
-----
Eddison Chen (KS '22 '24), Aarush Goradia (CO '24), Ethan Imanuel (NJ '24), Benjamin Jiang (FL '23 '24), Rayoon Kim (PA '23 '24), Jason Lee (NC '23 '24), Puranjay Madupu (AZ '23 '24), Andy Mo (OH '23 '24), George Paret (FL '24), Arjun Raman (IN '24), Vincent Wang (TX '24), Channing Yang (TX '23 '24), and Jefferson Zhou (MN '23 '24) present:



[center]IMAGE[/center]

[center]Image credits to Simon Joeng.[/center]

2024 MATHCOUNTS Nationals alumni from all across the nation have come together to administer the first-ever ELMOCOUNTS Competition, a mock written by the 2024 Nationals alumni given to the 2025 Nationals participants. By providing the next generation of mathletes with free, high quality practice, we're here to boast how strong of an alumni community MATHCOUNTS has, as well as foster interest in the beautiful art that is problem writing!

The tests and their corresponding submissions forms will be released here, on this thread, on Monday, April 21, 2025. The deadline is May 10, 2025. Tests can be administered asynchronously at your home or school, and your answers should be submitted to the corresponding submission form. If you include your AoPS username in your submission, you will be granted access to the private discussion forum on AoPS, where you can discuss the tests even before the deadline.
[list=disc]
[*] "How do I know these tests are worth my time?" [/*]
[*] "Who can participate?" [/*]
[*] "How do I sign up?" [/*]
[*] "What if I have multiple students?" [/*]
[*] "What if a problem is ambiguous, incorrect, etc.?" [/*]
[*] "Will there be solutions?" [/*]
[*] "Will there be a Countdown Round administered?" [/*]
[/list]
If you have any other questions, feel free to email us at elmocounts2025@gmail.com (or PM me)!
106 replies
1 viewing
vincentwant
Apr 20, 2025
mathkiddus
an hour ago
Combo problem
soryn   3
N 2 hours ago by soryn
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
3 replies
soryn
Yesterday at 6:33 AM
soryn
2 hours ago
Looking for the smallest ghost
Justpassingby   5
N 2 hours ago by venhancefan777
Source: 2021 Mexico Center Zone Regional Olympiad, problem 1
Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$, and for $n\ge p$, $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$. We say that a positive integer is a ghost if it doesn’t appear in $S$.
What is the smallest ghost that is not a multiple of $p$?

Proposed by Guerrero
5 replies
Justpassingby
Jan 17, 2022
venhancefan777
2 hours ago
non-symmetric ineq (for girls)
easternlatincup   36
N 2 hours ago by Tony_stark0094
Source: Chinese Girl's MO 2007
For $ a,b,c\geq 0$ with $ a+b+c=1$, prove that

$ \sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\leq \sqrt{3}$
36 replies
easternlatincup
Dec 30, 2007
Tony_stark0094
2 hours ago
Turbo's en route to visit each cell of the board
Lukaluce   20
N 2 hours ago by Mathgloggers
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
20 replies
Lukaluce
Apr 14, 2025
Mathgloggers
2 hours ago
Divisibility on 101 integers
BR1F1SZ   3
N 2 hours ago by ClassyPeach
Source: Argentina Cono Sur TST 2024 P2
There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.
3 replies
BR1F1SZ
Aug 9, 2024
ClassyPeach
2 hours ago
BMO 2021 problem 3
VicKmath7   19
N 2 hours ago by NuMBeRaToRiC
Source: Balkan MO 2021 P3
Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite.

Proposed by Serbia
19 replies
VicKmath7
Sep 8, 2021
NuMBeRaToRiC
2 hours ago
USAMO 2002 Problem 4
MithsApprentice   89
N 3 hours ago by blueprimes
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y)  \] for all pairs of real numbers $x$ and $y$.
89 replies
MithsApprentice
Sep 30, 2005
blueprimes
3 hours ago
pqr/uvw convert
Nguyenhuyen_AG   8
N 4 hours ago by Victoria_Discalceata1
Source: https://github.com/nguyenhuyenag/pqr_convert
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression $f(a,b,c)$ into $f(p,q,r)$ or $f(u,v,w)$ can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!

Download: pqr_convert

Screenshot:
IMAGE
IMAGE
8 replies
Nguyenhuyen_AG
Apr 19, 2025
Victoria_Discalceata1
4 hours ago
Inspired by hlminh
sqing   2
N 4 hours ago by SPQ
Source: Own
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that $$ |a-kb|+|b-kc|+|c-ka|\leq \sqrt{3k^2+2k+3}$$Where $ k\geq 0 . $
2 replies
sqing
Yesterday at 4:43 AM
SPQ
4 hours ago
A cyclic inequality
KhuongTrang   3
N 4 hours ago by KhuongTrang
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
3 replies
KhuongTrang
Monday at 4:18 PM
KhuongTrang
4 hours ago
Life, the universe and everyone sillies
Squidget   13
N Nov 13, 2024 by giratina3
Source: 2024 AMC 10B Problem #8
Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?

$
\textbf{(A) }0 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }4 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad
$
13 replies
Squidget
Nov 13, 2024
giratina3
Nov 13, 2024
Life, the universe and everyone sillies
G H J
G H BBookmark kLocked kLocked NReply
Source: 2024 AMC 10B Problem #8
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Squidget
433 posts
#1
Y by
Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?

$
\textbf{(A) }0 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }4 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad
$
This post has been edited 4 times. Last edited by Eternica, Mar 7, 2025, 5:05 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HungryCalculator
539 posts
#2
Y by
D for sure
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alexanderhamilton124
389 posts
#3 • 2 Y
Y by ranu540, Tem8
D cause (42)^4 = 6 mod 10
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
gracemoon124
872 posts
#4
Y by
(D) because there are 8 divisors of 42 total, so we pair up those divisors -> (42)^4 leads to (D)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
pingpongmerrily
3568 posts
#5
Y by
D confirmed
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
paganiniana
211 posts
#6
Y by
bash mod 10 so D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
megarnie
5590 posts
#7
Y by
Quick: Note that each prime is counted four times, so the answer is an even $4$th power, implying $\boxed{\textbf{(D)}\ 6}$.
This post has been edited 2 times. Last edited by megarnie, Nov 13, 2024, 5:23 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Tem8
238 posts
#8
Y by
All divisors of 42 are:
\begin{align*}
1\cdot42&=42\\
2\cdot21&=42\\
3\cdot14&=42\\
6\cdot7&=42.
\end{align*}Thus, the units digit of $B$ is $42^4 \equiv 2^4 = 16 \equiv \boxed{\textbf{(D) } 6} \pmod{10}$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
idk12345678
381 posts
#9
Y by
I just did 42^4 since d(42)/2 = 4
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MathRook7817
659 posts
#10
Y by
42^(8/2)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
andrewcheng
525 posts
#11
Y by
WAIT I READ THE QUESTION WRONG AND ADDED THE SUM OF THE DIVISORS AND STILL GOT IT CORRECT
This post has been edited 1 time. Last edited by andrewcheng, Nov 13, 2024, 6:36 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
eg4334
631 posts
#12
Y by
$42^{\frac{d(n)}{2}} = 42^4 \equiv \boxed{6} \pmod{10}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
elizhang101412
1204 posts
#13
Y by
i completely forgot you could pair them up and just multiplied manually :skull:
still got it right though
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
giratina3
494 posts
#14
Y by
You bash all the divisors of 42 and multiply the units digits
Z K Y
N Quick Reply
G
H
=
a