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

Stay ahead of learning milestones! Enroll in a class over the summer!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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
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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
J
H
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
J
H
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
J
H
THREE People Meet at the SAME. TIME.
LilKirb   7
N 35 minutes ago by hellohi321
Three people arrive at the same place independently, at a random between $8:00$ and $9:00.$ If each person remains there for $20$ minutes, what's the probability that all three people meet each other?

I'm already familiar with the variant where there are only two people, where you Click to reveal hidden text It was an AIME problem from the 90s I believe. However, I don't know how one could visualize this in a Click to reveal hidden text Help on what to do?
7 replies
LilKirb
Yesterday at 1:06 PM
hellohi321
35 minutes ago
J
H
Good Permutations in Modulo n
swynca   11
N an hour ago by Assassino9931
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
11 replies
swynca
Apr 27, 2025
Assassino9931
an hour ago
J
H
Israel Number Theory
mathisreaI   67
N an hour ago by lolsamo
Source: IMO 2022 Problem 5
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
67 replies
mathisreaI
Jul 13, 2022
lolsamo
an hour ago
J
H
IMO 2016 Shortlist, N6
dangerousliri   69
N an hour ago by MR.1
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.

Proposed by Dorlir Ahmeti, Albania
69 replies
dangerousliri
Jul 19, 2017
MR.1
an hour ago
J
H
Number of Solutions is 2
Miku3D   30
N 2 hours ago by lakshya2009
Source: 2021 APMO P1
Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.
30 replies
Miku3D
Jun 9, 2021
lakshya2009
2 hours ago
J
H
Problem 3
blug   3
N 2 hours ago by sunken rock
Source: Czech-Polish-Slovak Junior Match 2025 Problem 3
In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that
$$AE+BD=KL.$$
3 replies
blug
Yesterday at 4:47 PM
sunken rock
2 hours ago
J
H
Computing functions
BBNoDollar   6
N 2 hours ago by BBNoDollar
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
6 replies
BBNoDollar
May 18, 2025
BBNoDollar
2 hours ago
J
H
Hard Inequality
danilorj   5
N 3 hours ago by danilorj
Let $a, b, c > 0$ with $a + b + c = 1$. Prove that:
\[ \sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3}, \]with equality if and only if $a = b = c = \frac{1}{3}$.
5 replies
danilorj
Today at 5:17 AM
danilorj
3 hours ago
J
H
easy geo
ErTeeEs06   5
N 3 hours ago by Adywastaken
Source: BxMO 2025 P3
Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.
5 replies
ErTeeEs06
Apr 26, 2025
Adywastaken
3 hours ago
J
H
Inspired by SXJX (12)2022 Q1167
sqing   4
N 3 hours ago by sqing
Source: Own
Let $ a,b,c>0 $. Prove that$$\frac{kabc-1} {abc(a+b+c+8(2k-1))}\leq \frac{1}{16 }$$Where $ k>\frac{1}{2}.$
4 replies
sqing
Yesterday at 4:01 AM
sqing
3 hours ago
J
H
Geometry hard problem.
noneofyou34   3
N 3 hours ago by noneofyou34
In a circle of radius R, three chords of length R are given. Their ends are joined with segments to
obtain a hexagon inscribed in the circle. Show that the midpoints of the new chords are the vertices of
an equilateral triang
3 replies
noneofyou34
Today at 9:50 AM
noneofyou34
3 hours ago
J
H
Quite straightforward
steven_zhang123   1
N 3 hours ago by Mathzeus1024
Given that the sequence $\left \{ a_{n} \right \} $ is an arithmetic sequence, $a_{1}=1$, $a_{2}+a_{3}+\dots+a_{10}=144$. Let the general term $b_{n}$ of the sequence $\left \{ b_{n} \right \}$ be $\log_{a}{(1+\frac{1}{a_{n}} )} ( a > 0 \text{and} a \ne 1)$, and let $S_{n}$ be the sum of the $n$ terms of the sequence $\left \{ b_{n} \right \}$. Compare the size of $S_{n}$ with $\frac{1}{3} \log_{a}{(1+\frac{1}{a_{n}} )} $.
1 reply
steven_zhang123
Jan 11, 2025
Mathzeus1024
3 hours ago
J
H
Inequalities
sqing   12
N 3 hours ago by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
12 replies
sqing
May 13, 2025
sqing
3 hours ago
J
H
Inequalities
sqing   0
4 hours ago
Let $ a,b,c>0. $ Prove that$$a^2+b^2+c^2+abc-k(a+b+c)\geq 3k+2-2(k+1)\sqrt{k+1}$$Where $7\geq k \in N^+.$
$$a^2+b^2+c^2+abc-3(a+b+c)\geq-5$$
0 replies
sqing
4 hours ago
0 replies
J
H
J
H
Binomial Coefficient Stuff
aie8920   4
N Mar 20, 2021 by fungarwai
For positive integers $n \geq k,$ show that
\[\frac{\binom{n+1}{k+1} \cdot \binom{n}{k}}{n-k+1}\]is an integer.
Idk, this problem feels easy, but I'm not getting anywhere so far.
4 replies
aie8920
Mar 18, 2021
fungarwai
Mar 20, 2021
J
Binomial Coefficient Stuff
G H J
Reveal topic tags
binomial coefficients
L
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.
aie8920
1270 posts
aie8920
#1 h  Mar 18, 2021, 4:48 AM
Y by
For positive integers $n \geq k,$ show that
\[\frac{\binom{n+1}{k+1} \cdot \binom{n}{k}}{n-k+1}\]is an integer.
Idk, this problem feels easy, but I'm not getting anywhere so far.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SilverDragon246
678 posts
SilverDragon246
#2 h  Mar 18, 2021, 5:41 AM
Y by
Easiest is to just expand and try to show its an integer, alternatively, there seems like there is probably a combinatorial interpretation of this
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
scrabbler94
7554 posts
scrabbler94
#3 h  Mar 18, 2021, 5:48 AM • 2 Y
Y by aie8920, fungarwai
$\frac{\binom{n}{k}}{n-k+1} = \frac{n!}{(n-k+1)!k!} = \frac{1}{n+1}\binom{n+1}{k}$ so the problem reduces to showing $\binom{n+1}{k} \binom{n+1}{k+1}$ is divisible by $n+1$ for all $n \ge k$. I'd imagine this is a well-known result or has an easier combinatorial interpretation...
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Billybillybobjoejr.
352 posts
Billybillybobjoejr.
#4 h  Mar 18, 2021, 6:01 AM • 3 Y
Y by Mango247, Mango247, Mango247
i think there should be a nice proof using p-adic valuation on the prime factors of n+1 while using the property that gcd(k, k+1) = 1
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
fungarwai
865 posts
fungarwai
#5 h  Mar 20, 2021, 11:47 AM • 1 Y
Y by aie8920
$\nu_p((n+1)!n!)\ge \nu_p((k+1)!(n-k)!k!(n+1-k)!)$ caused by $(k,k+1)=1$

I prefer noting $n \mid \binom{n}{m}\binom{n}{m+1}$ on my blog as #3 suggested

Proof
Z K Y
N Quick Reply
G
H
=
a