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
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
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
Problem in probability theory
Tip_pay   1
N 3 minutes ago by Tip_pay
Find the probability that if four numbers from $1$ to $100$ (inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by $3$, or all will be divisible by $5$
1 reply
Tip_pay
Yesterday at 11:00 AM
Tip_pay
3 minutes ago
J
H
Square and equilateral triangle
m4thbl3nd3r   1
N 4 minutes ago by mpcnotnpc
Let $ABCD$ be a square and a point $X$ lies on the interior of $ABCD$ such that triangle $BDX$ is equilateral. Evaluate $\angle AXD$
1 reply
m4thbl3nd3r
7 minutes ago
mpcnotnpc
4 minutes ago
J
H
NT Problem
tiendat004   0
4 minutes ago
Let $a\in\mathbb{N}_+$ with $a$ is coprime to $21$. It is known that for every $s\in\mathbb{N}_+$, there always exist $r,t\in\mathbb{N}_+$ satisfying $$a+7s^3=r^3+7t^3.$$Prove that $a$ is a cube.
0 replies
tiendat004
4 minutes ago
0 replies
J
H
series and factorials?
jenishmalla   7
N 26 minutes ago by mpcnotnpc
Source: 2025 Nepal ptst p4 of 4
Find all pairs of positive integers \( n \) and \( x \) such that
\[ 1^n + 2^n + 3^n + \cdots + n^n = x! \]
(Petko Lazarov, Bulgaria)
7 replies
jenishmalla
Mar 15, 2025
mpcnotnpc
26 minutes ago
J
H
A board with crosses that we color
nAalniaOMliO   1
N an hour ago by EmersonSoriano
Source: Belarusian National Olympiad 2025
In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses.
Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
1 reply
nAalniaOMliO
Mar 28, 2025
EmersonSoriano
an hour ago
J
H
Inspired by JK1603JK
sqing   1
N 2 hours ago by Soupboy0
Source: Own
Let $ a,b,c\geq 0 $ and $ab+bc+ca=1.$ Prove that$$\frac{abc-2}{abc-1}\ge \frac{4(a^2b+b^2c+c^2a)}{a^3b+b^3c+c^3a+1} $$$$\frac{abc-1}{abc-2}\ge \frac{(\sqrt 2-1)(a^2b+b^2c+c^2a+1)}{a^3b+b^3c+c^3a+1} $$
1 reply
sqing
2 hours ago
Soupboy0
2 hours ago
J
H
Olympiad Geometry problem-second time posting
kjhgyuio   7
N 2 hours ago by kjhgyuio
Source: smo problem
In trapezium ABCD,AD is parallel to BC and points E and F are midpoints of AB and DC respectively. If
Area of AEFD/Area of EBCF =√3 + 1/3-√3 and the area of triangle ABD is √3 .find the area of trapezium ABCD
7 replies
kjhgyuio
Apr 2, 2025
kjhgyuio
2 hours ago
J
H
Geometry problem-second time posting
kjhgyuio   0
2 hours ago
Source: smo roudn 2

A square is cut into several rectangles, none of which is a square ,so that the sides of each rectangles are parallel to the sides of a square .For each rectangle with sides a,b,a<b compute the ratio a/b Prove that the sum of these ratios is at least 1
0 replies
kjhgyuio
2 hours ago
0 replies
J
H
Proving ZA=ZB
nAalniaOMliO   5
N 2 hours ago by EmersonSoriano
Source: Belarusian National Olympiad 2025
Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$.
Prove that $ZA=ZH$.
Vadzim Kamianetski
5 replies
nAalniaOMliO
Mar 28, 2025
EmersonSoriano
2 hours ago
J
H
Inequality from China
sqing   3
N 2 hours ago by sqing
Source: lemondian(https://kuing.cjhb.site/thread-13667-1-1.html)
Let $x\in (0,\frac{\pi}{2}) . $ Prove that $$tanx\ge x^k$$Where $ k=1,2,3,4.$
3 replies
1 viewing
sqing
Yesterday at 1:11 PM
sqing
2 hours ago
J
H
NMO (Nepal) Problem 4
khan.academy   8
N 2 hours ago by godchunguus
Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer.

Proposed by Prajit Adhikari, Nepal
8 replies
khan.academy
Mar 17, 2024
godchunguus
2 hours ago
J
H
2019 Nepal National Mathematics Olympiad
Piinfinity   3
N 3 hours ago by godchunguus
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
3 replies
Piinfinity
Oct 13, 2020
godchunguus
3 hours ago
J
H
pretty well known
dotscom26   2
N 3 hours ago by Giant_PT
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

2 replies
dotscom26
Yesterday at 2:03 AM
Giant_PT
3 hours ago
J
H
Unsolved NT, 3rd time posting
GreekIdiot   10
N 3 hours ago by mathprodigy2011
Source: own
Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
10 replies
GreekIdiot
Mar 26, 2025
mathprodigy2011
3 hours ago
J
H
J
H
limit of a sequence -hard
XXX1994   3
N Apr 4, 2016 by hakaselp
Define $(x_n)$ with $x_1=x_2=1$ and $x_{n+2}=x_{n+1}^2-\frac{x_n}{2}$.Find $\lim x_n$
3 replies
XXX1994
Apr 1, 2016
hakaselp
Apr 4, 2016
J
limit of a sequence -hard
G H J
Reveal topic tags
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.
XXX1994
135 posts
XXX1994
#1 h  Apr 1, 2016, 3:11 PM • 2 Y
Y by Adventure10, Mango247
Define $(x_n)$ with $x_1=x_2=1$ and $x_{n+2}=x_{n+1}^2-\frac{x_n}{2}$.Find $\lim x_n$
This post has been edited 1 time. Last edited by XXX1994, Apr 3, 2016, 8:54 AM
Reason: Title
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
XXX1994
135 posts
XXX1994
#2 h  Apr 2, 2016, 11:54 AM • 2 Y
Y by Adventure10, Mango247
Anyone help me,please ? >.> I try to use 2 subsequeces $x_{2k}$ and $x_{2k-1}$ but it doesn't work or could we use $y_{n+1}=y_{n}^2-\frac{y_{n}}{2};y_1=1$ or anything like that ?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
XXX1994
135 posts
XXX1994
#3 h  Apr 4, 2016, 7:40 AM • 1 Y
Y by Adventure10
Anyone ? Also I want to ask if we have a method for this type of limit $x_{n+2}=f(x_{n+1};x_n)$.Thank you :)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hakaselp
3 posts
hakaselp
#4 h  Apr 4, 2016, 3:59 PM • 2 Y
Y by Adventure10, Mango247
i think i got the answer..
(sol)just calculate..and then you got x1=1, x2=1, x3=1/2, x4=-1/4, x5=-3/16
lemma1:"n>_(equal involved)4, [xn](this is not gauss..absolute symbol..([-3]=3, [4]=4))<_1/4"
(pf)suppose 4<_n<_M => [xn]<_1/4(M>_5)
[x(n+2)]=[x(n+1)^2-xn/2]<_[x(n+1)^2]+[xn/2]<_1/16+1/8<1/4
so u can get 4<_n<_M+1=> [xn]<_1/4
so repeat this process and u can get 4<_n =>[xn]<_1/4
so beacause "x4=-1/4, x5=-3/16", lemma1 is proved..

lemma2:y(n+1)=yn^2+yn/2 & 0<y1<1/2 =>lim yn =0
(pf)if 0<yn<1/2, 0<y(n+1)=yn(yn+1/2)<yn.. and 0<y1<1/2 so about all of n, 0<yn<1/2
and so, 0<y(n+1)<yn..and so, 0<y(n+1)=(yn+1/2)yn<(y1+1/2)yn=pyn(let y1+1/2=p)
because 0<y1<1/2, 0<p<1 so lim yn=0

let y1=1/4
n>_4 => [xn]<y1
[x(n+2)]=[x(n+1)^2-xn/2]<_[x(n+1)^2]+[xn/2]<_y1^2+yn/2=y2
so n>_6 => [xn]<_y2
now you now easly that "about all of m, special N is exsist so that n>_N =>[xn]<_ym"
so u can show lim[xn]=0 using "caushy's defenition" and because -[xn]<_xn<_[xn], limxn=0 ..Q.E.D..

am i right? please read mt solution..

(i am poor at grammer because i'm not native..sorry..)
This post has been edited 3 times. Last edited by hakaselp, Apr 5, 2016, 3:32 AM
Reason: to perfect solution
Z K Y
N Quick Reply
G
H
=
a