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 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
Some Problems
hashbrown2009   0
10 minutes ago
1. If x squared minus (x squared times pi)/4 equals 3+ 3 root 3 -2pi, solve for x.
2. You're playing a game where you need to roll at least N to win. You can either roll 2 6-sided fair dice or a 12-sided fair die. For what value of n between 1 and 12, inclusive, is the probability of winning equal no matter which dice you choose?
3. Define the polynomial P(x)=(x^3)+1. Find the sum of the coefficients of P(P(P(x))).
4. Jerry chooses 2 positive integers x and y that yield a product of 6^5, or 7776. He then computes the value of n, which is the GCD of (a,b). How many possible values of n can Jerry get?

Provide solutions to answers


Also how hard are these problems: AMC8, AMC10, MathCounts, or AMC12?
0 replies
hashbrown2009
10 minutes ago
0 replies
J
H
FTW tournament!
evt917   343
N 14 minutes ago by sadas123
[center]Since all FTW tournaments have dramatically failed, I'm trying a different format. Here is how it works:

1. Type \signup{your rating (type 800 for unrated)}

2. You will pick who you want to play with. You can play if they accept your challenge. So basically the players run everything. Just don't intentionally play low-rated people. Also try to play different people so everyone gets a chance to play! ONLY two player games.

3. If you win, you get 2 points. Ties get one point, and losses get zero.

4. I do not know everybody's time preferences. Because so, I will announce in advance which two players will be playing, so they themselves can organize a game themselves. Remember, THE PLAYERS ARE ORGANIZING THE GAMES THEMSELVES!!! The format is up to them, but please make the time control at least 20 seconds. Please announce the results of the game here so i can update the scoreboard. Games can be unrated.

recommended format if you cannot decide



5. The tournament goes on until april 10th! Extremely long, right? Note that you can still signup after the first games has started, but you will have a disadvantage because some people who signed up as soon as the tournament started already has points.

6. Once you are done with your game, you can find a new opponent and play with them if they want. Note that you must play opponents within the tournament. If you play in the tournament, you are automatically signed up. Have fun!


[rule]

Questions and Answers

All signups and ratings

[rule]

LIVE LEADERBOARD:

1st place: 47 points | 17W 3L 3T | Yrock
2nd place: 14 points | 6W 3L 2T | jb2015007
3rd place: 5 points | 2W 8L 1T | sadas123

4th place: 4 points | 1W 2L 0T | IcyFire500
5th place: 0 points | 0W 1L 0T | NS0004
343 replies
evt917
Apr 3, 2025
sadas123
14 minutes ago
J
H
mathcounts state score thread
Soupboy0   75
N 18 minutes ago by iwastedmyusername
\begin{table}[]
\begin{tabular}{llllll} Username & Score & Sprint & Target & Nats? & Sillies \\ Soupboy0 & 40 & 24 & 16 & yes & 6 \\ & & & & & \\ & & & & & \end{tabular}\end{table}
75 replies
+1 w
Soupboy0
Apr 1, 2025
iwastedmyusername
18 minutes ago
J
H
The daily problem!
Leeoz   65
N 44 minutes ago by HacheB2031
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]

Past Problems!
65 replies
Leeoz
Mar 21, 2025
HacheB2031
44 minutes ago
J
H
Two Functional Inequalities
Mathdreams   6
N an hour ago by Assassino9931
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
6 replies
Mathdreams
Today at 1:34 PM
Assassino9931
an hour ago
J
H
Pythagorean new journey
XAN4   2
N an hour ago by mathprodigy2011
Source: Inspired by sarjinius
The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$, if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.
2 replies
XAN4
Today at 3:41 AM
mathprodigy2011
an hour ago
J
H
sqrt(2) and sqrt(3) differ in at least 1000 digits
Stuttgarden   2
N an hour ago by straight
Source: Spain MO 2025 P3
We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.
2 replies
Stuttgarden
Mar 31, 2025
straight
an hour ago
J
H
combinatorics and number theory beautiful problem
Medjl   2
N an hour ago by mathprodigy2011
Source: Netherlands TST for BxMo 2017 problem 4
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$
2 replies
Medjl
Feb 1, 2018
mathprodigy2011
an hour ago
J
H
Squence problem
AlephG_64   1
N 2 hours ago by RagvaloD
Source: 2025 Finals Portuguese Math Olympiad P1
Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?
1 reply
1 viewing
AlephG_64
Yesterday at 1:19 PM
RagvaloD
2 hours ago
J
H
50 points in plane
pohoatza   12
N 2 hours ago by de-Kirschbaum
Source: JBMO 2007, Bulgaria, problem 3
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
12 replies
pohoatza
Jun 28, 2007
de-Kirschbaum
2 hours ago
J
H
beautiful functional equation problem
Medjl   6
N 2 hours ago by Sadigly
Source: Netherlands TST for BxMO 2017 problem 2
Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$$f(p)=1$ for all prime numbers $p$.
$ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$
6 replies
Medjl
Feb 1, 2018
Sadigly
2 hours ago
J
H
Line Combining Fermat Point, Orthocenter, and Centroid
cooljoseph   0
2 hours ago
On triangle $ABC$, draw exterior equilateral triangles on sides $AB$ and $AC$ to obtain $ABC'$ and $ACB'$, respectively. Let $X$ be the intersection of the altitude through $B$ and the median through $C$. Let $Y$ be the intersection of the altitude through $A$ and line $CC'$. Let $Z$ be the intersection of the median through $A$ and the line $BB'$. Prove that $X$, $Y$, and $Z$ lie on a common line.

IMAGE
0 replies
cooljoseph
2 hours ago
0 replies
J
H
complete integral values
Medjl   2
N 2 hours ago by Sadigly
Source: Netherlands TST for BxMO 2017 problem 1
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
2 replies
Medjl
Feb 1, 2018
Sadigly
2 hours ago
J
H
interesting ineq
nikiiiita   5
N 2 hours ago by nikiiiita
Source: Own
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
5 replies
nikiiiita
Jan 29, 2025
nikiiiita
2 hours ago
J
H
J
H
Quadratics problem I came up with
V0305   3
N Yesterday at 12:59 AM by jb2015007
$P(x)$ is a quadratic polynomial such that $P(1) = 2$ and $P(2) = 5$. Find all possible values of $P(7) - P(-4)$.
3 replies
V0305
Nov 3, 2024
jb2015007
Yesterday at 12:59 AM
J
Quadratics problem I came up with
G H J
Reveal topic tags
quadratics
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.
V0305
662 posts
V0305
#1 h  Nov 3, 2024, 11:10 PM
Y by
$P(x)$ is a quadratic polynomial such that $P(1) = 2$ and $P(2) = 5$. Find all possible values of $P(7) - P(-4)$.
This post has been edited 2 times. Last edited by V0305, Nov 3, 2024, 11:26 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Rabbit47
780 posts
Rabbit47
#2 h  Nov 4, 2024, 1:48 AM • 2 Y
Y by aidan0626, HockeyMaster85
We represent $P(x)$ as $ax^2+bx+c$. From the first equation, we have $a+b+c=2$ and from the second we have $4a+2b+c=5$. Note we are trying to find, $49a+7b-16a+4b-c$ which simplifies to $33a+11b$. We also have from the first two equations $3a+b=3$, and multiplying on both sides by $11$ we have $33a+11b=\boxed{33}$, which is our answer.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Dixit1
10 posts
Dixit1
#3 h  Apr 4, 2025, 4:56 AM
Y by
Since p(1)=2 and p(2)=5, we have the equations

a+b+c=2 and 4a+2b+c=5
Equating both, we will get
b=3(1-a) and c=2a-1
Then p(7) would be 30a+20 and p(-4) would be 30a-13. Subtracting both the equations, we get the answer to be 33.

Hence the answer is 33
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jb2015007
1809 posts
jb2015007
#4 h  Yesterday at 12:59 AM
Y by
Sol

Also hi V0305!
Z K Y
N Quick Reply
G
H
=
a