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

Join our FREE webinar on May 1st to learn about managing anxiety.
No tags match your search
M
G
Topic
First Poster
Last Poster
H
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
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
RMM 2013 Problem 1
dr_Civot   31
N 24 minutes ago by cursed_tangent1434
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
31 replies
dr_Civot
Mar 2, 2013
cursed_tangent1434
24 minutes ago
J
H
Inspired by old results
sqing   0
24 minutes ago
Source: Own
Let $ a , b , c>0 $and $ abc=1 $. Prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3 \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$h
0 replies
sqing
24 minutes ago
0 replies
J
H
amazing balkan combi
egxa   7
N 26 minutes ago by Assassino9931
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
7 replies
egxa
Apr 27, 2025
Assassino9931
26 minutes ago
J
H
Question on Balkan SL
Fmimch   2
N 28 minutes ago by Assassino9931
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
2 replies
Fmimch
Today at 12:13 AM
Assassino9931
28 minutes ago
J
H
Or statement function
ItzsleepyXD   1
N 30 minutes ago by Haris1
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
1 reply
ItzsleepyXD
2 hours ago
Haris1
30 minutes ago
J
H
Inequlities
sqing   32
N 31 minutes ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
32 replies
sqing
Jul 19, 2024
sqing
31 minutes ago
J
H
Inequalities
sqing   2
N 35 minutes ago by sqing
Let $a,b,c> 0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1.$ Prove that
$$ (1-abc) (1-a)(1-b)(1-c) \ge 208 $$$$ (1+abc) (1-a)(1-b)(1-c) \le -224 $$$$(1+a^2b^2c^2) (1-a)(1-b)(1-c) \le -5840 $$
2 replies
sqing
Jul 12, 2024
sqing
35 minutes ago
J
H
Add a digit to obtain a new perfect square
Lukaluce   2
N 41 minutes ago by TopGbulliedU
Source: 2024 Junior Macedonian Mathematical Olympiad P4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.

Proposed by Ilija Jovčeski
2 replies
Lukaluce
Apr 14, 2025
TopGbulliedU
41 minutes ago
J
H
Simple inequality
sqing   7
N an hour ago by sqing
Source: Daniel Sitaru
Let $a,b,c>0$ . Prove that$$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$$
7 replies
sqing
Feb 10, 2017
sqing
an hour ago
J
H
Vector Vortex
steven_zhang123   1
N an hour ago by Mathzeus1024
Source: NS Issue 1 P3 (2014.4)
Let $v_{1}, v_{2}, \cdots, v_{n}$ be $n$ unit vectors on a plane, where $n$ is an odd number. Prove that there exist $\varepsilon _i\in \left \{ -1,1 \right \} $ for $i=1,2,\cdots,n$ such that $\left | \sum_{i=1}^{n} \varepsilon_i v_i \right | \le 1.$
1 reply
steven_zhang123
Feb 15, 2025
Mathzeus1024
an hour ago
J
H
China Northern MO 2009 p4 CNMO
parkjungmin   0
an hour ago
Source: China Northern MO 2009 p4 CNMO P4
The problem is too difficult.
0 replies
parkjungmin
an hour ago
0 replies
J
H
The Appetizer of Iran NT2023
alinazarboland   6
N an hour ago by A22-
Source: Iran MO 3rd round 2023 NT exam , P1
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
6 replies
alinazarboland
Aug 17, 2023
A22-
an hour ago
J
H
Basic geometry
AlexCenteno2007   5
N 4 hours ago by mathafou
Given an isosceles triangle ABC with AB=BC, the inner bisector of Angle BAC And cut next to it BC in D. A point E is such that AE=DC. The inner bisector of the AED angle cuts to the AB side at the point F. Prove that the angle AFE= angle DFE
5 replies
AlexCenteno2007
Feb 9, 2025
mathafou
4 hours ago
J
H
Generating Functions
greenplanet2050   6
N 4 hours ago by ohiorizzler1434
So im learning generating functions and i dont really understand why $1+2x+3x^2+4x^3+5x^4+…=\dfrac{1}{(1-x)^2}$

can someone help

thank you :)
6 replies
greenplanet2050
Yesterday at 10:42 PM
ohiorizzler1434
4 hours ago
J
H
J
H
how many quadrilaterals ?
Ecrin_eren   8
N Apr 16, 2025 by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
8 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Apr 16, 2025
J
how many quadrilaterals ?
G H J
Reveal topic tags
combinatorics
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.
Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 13, 2025, 4:01 PM • 1 Y
Y by Kizaruno
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathprodigy2011
324 posts
mathprodigy2011
#2 h  Apr 14, 2025, 2:54 AM • 2 Y
Y by yodaboss1110, Kizaruno
Ecrin_eren wrote:
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"

wait so basically your asking for the number of quadrilaterals that share all vertices with the 11-gon and all sides are diagonals?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ecrin_eren
56 posts
Ecrin_eren
#3 h  Apr 14, 2025, 5:15 AM
Y by
Yes exactly
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ecrin_eren
56 posts
Ecrin_eren
#4 h  Apr 14, 2025, 2:00 PM • 1 Y
Y by Kizaruno
Any ideas ?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathprodigy2011
324 posts
mathprodigy2011
#5 h  Apr 14, 2025, 7:59 PM • 2 Y
Y by Ecrin_eren, Kizaruno
Ecrin_eren wrote:
Any ideas ?
PIE could work nicely. 11 choose 4 total ways. Then count the number of quadrilaterals with at least 1 side shared and count with PIE. I'll write a full solution later since im at school rn. Stars and bars could also be good lol
This post has been edited 1 time. Last edited by mathprodigy2011, Apr 15, 2025, 5:32 PM
Reason: .
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ecrin_eren
56 posts
Ecrin_eren
#6 h  Apr 15, 2025, 5:18 PM
Y by
@mathprodigy could you please give your solution
This post has been edited 1 time. Last edited by Ecrin_eren, Apr 15, 2025, 5:18 PM
Reason: Mb
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathprodigy2011
324 posts
mathprodigy2011
#7 h  Apr 15, 2025, 5:31 PM
Y by
Ecrin_eren wrote:
@mathprodigy could you please give your solution

Click to reveal hidden text Don't think im right after doing this :skull:, No latex cuz im still at school
This post has been edited 1 time. Last edited by mathprodigy2011, Apr 15, 2025, 5:32 PM
Reason: school
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
alcumus_grinder_bad
1 post
alcumus_grinder_bad
#9 h  Apr 16, 2025, 3:51 PM • 1 Y
Y by Kizaruno
i could be super duper wrong becuase i am not very good at math but isnt this just the number of ways to choose 4 vertices out of a set of 11 such that each vertex isn't adjacent to another? in which case u could do a sort of stars and bars type approach I think.

Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathprodigy2011
324 posts
mathprodigy2011
#10 h  Apr 16, 2025, 5:27 PM • 1 Y
Y by Kizaruno
alcumus_grinder_bad wrote:
i could be super duper wrong becuase i am not very good at math but isnt this just the number of ways to choose 4 vertices out of a set of 11 such that each vertex isn't adjacent to another? in which case u could do a sort of stars and bars type approach I think.

Click to reveal hidden text

well you have to adjust for each vertex so i don't think this is right but im unsure
Z K Y
N Quick Reply
G
H
=
a