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

We have your learning goals covered with Spring and Summer courses available. Enroll today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/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, Mar 2 - Jun 22
Friday, Mar 28 - Jul 18
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
Tuesday, Mar 25 - Jul 8
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
Sunday, Mar 23 - Jul 20
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
Sunday, Mar 16 - Jun 8
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
Monday, Mar 17 - Jun 9
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
Sunday, Mar 2 - Jun 22
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
Tuesday, Mar 4 - Aug 12
Sunday, Mar 23 - Sep 21
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
Sunday, Mar 16 - Sep 14
Tuesday, Mar 25 - Sep 2
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
Sunday, Mar 23 - Aug 3
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
Sunday, Mar 16 - Aug 24
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
Wednesday, Mar 5 - May 21
Tuesday, Jun 10 - Aug 26

Calculus
Sunday, Mar 30 - Oct 5
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
Sunday, Mar 23 - Jun 15
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
Tuesday, Mar 4 - May 20
Monday, Mar 31 - Jun 23
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
Monday, Mar 24 - Jun 16
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
Sunday, Mar 30 - Jun 22
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Tuesday, Mar 25 - Sep 2
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
Mar 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
Existence of AP of interesting integers
DVDthe1st   33
N 4 minutes ago by tchange7575
Source: 2018 China TST Day 1 Q2
A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
33 replies
DVDthe1st
Jan 2, 2018
tchange7575
4 minutes ago
J
H
A diophantine equation
crazyfehmy   13
N 6 minutes ago by Primeniyazidayi
Source: Turkey Junior National Olympiad 2012 P1
Let $x, y$ be integers and $p$ be a prime for which

\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
13 replies
crazyfehmy
Dec 12, 2012
Primeniyazidayi
6 minutes ago
J
H
nf(f(n)) = f(n)^2, f : N->N
Zhero   19
N 6 minutes ago by HamstPan38825
Source: ELMO Shortlist 2010, A1; also ELMO #4
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.

Carl Lian and Brian Hamrick.
19 replies
Zhero
Jul 5, 2012
HamstPan38825
6 minutes ago
J
H
Number Theory
MuradSafarli   1
N 27 minutes ago by Amkan2022
Find all natural numbers \( a, b, c \) such that

\[ 2^a \cdot 3^b + 1 = 5^c. \]
1 reply
MuradSafarli
38 minutes ago
Amkan2022
27 minutes ago
J
H
RMM 2019 Problem 2
math90   77
N 29 minutes ago by ihatemath123
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
77 replies
math90
Feb 23, 2019
ihatemath123
29 minutes ago
J
H
Two circles concur on a line
math154   59
N 43 minutes ago by Mathandski
Source: ELMO Shortlist 2012, G1; also ELMO #1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.

Ray Li.
59 replies
math154
Jul 2, 2012
Mathandski
43 minutes ago
J
H
functional equation
Anni   7
N an hour ago by HamstPan38825
Source: albanian TST 2008 bmo
Find all functions $f: \mathbb R \to \mathbb R$ such that
\[ f(x+f(y))=y+f(x+1),\]for all $x,y \in \mathbb R$.
7 replies
Anni
May 24, 2009
HamstPan38825
an hour ago
J
H
"pseudo-Fibonnaci" sequence
pohoatza   11
N an hour ago by asdf334
Source: IMO Shortlist 2006, Algebra 3
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.

Remark: A sum over the elements of the empty set is assumed to be $0$.
11 replies
pohoatza
Jun 28, 2007
asdf334
an hour ago
J
H
IMO ShortList 2001, algebra problem 4
orl   35
N an hour ago by HamstPan38825
Source: IMO ShortList 2001, algebra problem 4
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.
35 replies
orl
Sep 30, 2004
HamstPan38825
an hour ago
J
H
Arithmetic Sequence of Products
GrantStar   18
N an hour ago by hgomamogh
Source: IMO Shortlist 2023 N4
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
18 replies
GrantStar
Jul 17, 2024
hgomamogh
an hour ago
J
H
sequence (.) eventually becomes constant.
N.T.TUAN   60
N an hour ago by shendrew7
Source: USAMO 2007
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
60 replies
N.T.TUAN
Apr 26, 2007
shendrew7
an hour ago
J
H
A line tangent to a circumcircle
ericxyzhu   6
N an hour ago by zhenghua
Source: Lusophon Mathematical Olympiad 2021 Problem 3
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$.

Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.
6 replies
ericxyzhu
Dec 19, 2021
zhenghua
an hour ago
J
H
Number theory
Ecrin_eren   1
N an hour ago by Ecrin_eren
Show that there are no prime numbers satisfying the equation

(p + r)^q + (q + r)^p = (p + q)^r.

1 reply
Ecrin_eren
Mar 15, 2025
Ecrin_eren
an hour ago
J
H
f(f(x)+y)+f(x+y)=2x+2f(y)
parmenides51   4
N an hour ago by jasperE3
Source: 2015 AGCN Competition p1 by bobthesmartypants https://artofproblemsolving.com/community/c5h1128876p5232794
Find all functions $f:\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ satisfying$$f(f(x)+y)+f(x+y)=2x+2f(y)$$
4 replies
parmenides51
Dec 5, 2023
jasperE3
an hour ago
J
H
J
H
Inspired by my own results
sqing   1
N Today at 9:08 AM by sqing
Source: Own
Let $ a , b $ be reals such that $ a+b+a^2+b^2=1. $ Show that$$ \frac{4(4-\sqrt3) }{13} \le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{4(4+\sqrt3) }{13} $$Let $ a , b\geq 0 $ and $ a+b+a^2+b^2=1. $ Show that$$ \frac{15+\sqrt 5 }{10} \le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{4(4+\sqrt3) }{13} $$
1 reply
sqing
Today at 8:50 AM
sqing
Today at 9:08 AM
J
Inspired by my own results
G H J
Reveal topic tags
inequalities proposed
algebra
L
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41050 posts
sqing
#1 h  Today at 8:50 AM
Y by
Let $ a , b $ be reals such that $ a+b+a^2+b^2=1. $ Show that$$ \frac{4(4-\sqrt3) }{13} \le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{4(4+\sqrt3) }{13} $$Let $ a , b\geq 0 $ and $ a+b+a^2+b^2=1. $ Show that$$ \frac{15+\sqrt 5 }{10} \le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{4(4+\sqrt3) }{13} $$
This post has been edited 1 time. Last edited by sqing, Today at 9:04 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
41050 posts
sqing
#2 h  Today at 9:08 AM
Y by
Let $ a , b\geq 0 $ and $ a+b+a^2+ab+b^2=1. $ Show that$$ \frac{15+\sqrt 5 }{10} \le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{9 }{5} $$Let $ a , b $ be reals such that $ a+b+a^2+ab+b^2=1. $ Show that$$ 0.893869...\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{9 }{5} $$
Z K Y
N Quick Reply
G
H
=
a