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
Range of ab + bc + ca
bamboozled   2
N 25 minutes ago by Quantum-Phantom
Let $(a^2+1)(b^2+1)(c^2+1) = 9$, where $a, b, c \in R$, then the number of integers in the range of $ab + bc + ca$ is __
2 replies
bamboozled
6 hours ago
Quantum-Phantom
25 minutes ago
J
H
inquequality
ngocthi0101   10
N an hour ago by MS_asdfgzxcvb
let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$
10 replies
ngocthi0101
Sep 26, 2014
MS_asdfgzxcvb
an hour ago
J
H
Functional Equation
AnhQuang_67   5
N an hour ago by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+yf(x), \forall x, y \in \mathbb{R} $$
5 replies
AnhQuang_67
Yesterday at 4:50 PM
jasperE3
an hour ago
J
H
Special line through antipodal
Phorphyrion   8
N an hour ago by optimusprime154
Source: 2025 Israel TST Test 1 P2
Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.
8 replies
Phorphyrion
Oct 28, 2024
optimusprime154
an hour ago
J
H
PoP+Parallel
Solilin   1
N an hour ago by sansgankrsngupta
Source: Titu Andreescu, Lemmas in Olympiad Geometry
Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
1 reply
1 viewing
Solilin
2 hours ago
sansgankrsngupta
an hour ago
J
H
gcd of coefficients of polynomial
QueenArwen   2
N an hour ago by AshAuktober
Source: 46th International Tournament of Towns, Senior O-Level P5, Spring 2025
Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.
2 replies
QueenArwen
Mar 11, 2025
AshAuktober
an hour ago
J
H
Another config geo with concurrent lines
a_507_bc   15
N 2 hours ago by E50
Source: BMO SL 2023 G5
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
15 replies
a_507_bc
May 3, 2024
E50
2 hours ago
J
H
Dwarves at the river
Experia   1
N 2 hours ago by Radin_
Source: Stage PreIMO 2018 - Italy
There are $100$ dwarves, whose weigths are $1,\,2,\dots,\,100\,\text{kg}$, who want to cross a river. They have a small boat which can lift at most $100\,\text{kg}$ each time without sinking. For each journey of the boat a non-empty subset of the dwarves to be taken to the other side is chosen and one of these dwarves is chosen as the $\emph{rower}$ for that journey. Since return journeys are counter-current, no dwarf is able to do the rower for more than one return journey. Is it possible for all the dwarves to reach the other side of the river?
1 reply
Experia
Apr 23, 2022
Radin_
2 hours ago
J
H
Regarding Maaths olympiad prepration
omega2007   4
N 2 hours ago by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
4 replies
omega2007
Yesterday at 3:13 PM
omega2007
2 hours ago
J
H
square root problem that involves geometry
kjhgyuio   2
N 3 hours ago by ND_
If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

2 replies
kjhgyuio
4 hours ago
ND_
3 hours ago
J
H
Assisted perpendicular chasing
sarjinius   5
N 4 hours ago by hukilau17
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
5 replies
sarjinius
Mar 9, 2025
hukilau17
4 hours ago
J
H
Tangent.
steven_zhang123   2
N 5 hours ago by AshAuktober
Source: China TST 2001 Quiz 6 P1
In \( \triangle ABC \) with \( AB > BC \), a tangent to the circumcircle of \( \triangle ABC \) at point \( B \) intersects the extension of \( AC \) at point \( D \). \( E \) is the midpoint of \( BD \), and \( AE \) intersects the circumcircle of \( \triangle ABC \) at \( F \). Prove that \( \angle CBF = \angle BDF \).
2 replies
steven_zhang123
Mar 23, 2025
AshAuktober
5 hours ago
J
H
IMO ShortList 1998, algebra problem 1
orl   37
N 5 hours ago by Marcus_Zhang
Source: IMO ShortList 1998, algebra problem 1
Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that

\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]
37 replies
orl
Oct 22, 2004
Marcus_Zhang
5 hours ago
J
H
Integer Coefficient Polynomial with order
MNJ2357   9
N 5 hours ago by v_Enhance
Source: 2019 Korea Winter Program Practice Test 1 Problem 3
Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.
9 replies
MNJ2357
Jan 12, 2019
v_Enhance
5 hours ago
J
H
J
H
rectangular parallelepiped whose volume = total area = sum of lengths
parmenides51   0
May 21, 2022
Source: 2018 1st Kharkiv Masters Tournament VIII-IX 4.1
Is there a rectangular parallelepiped whose numerical values of volume, total area, and sum of the lengths of all the edges coincide?
0 replies
parmenides51
May 21, 2022
0 replies
J
rectangular parallelepiped whose volume = total area = sum of lengths
G H J
Reveal topic tags
geometry
3D geometry
parallelepiped
Kharkiv Masters
L
G H BBookmark kLocked kLocked NReply
Source: 2018 1st Kharkiv Masters Tournament VIII-IX 4.1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
parmenides51
30629 posts
parmenides51
#1 h  May 21, 2022, 6:30 PM • 1 Y
Y by Mango247
Is there a rectangular parallelepiped whose numerical values of volume, total area, and sum of the lengths of all the edges coincide?
Z K Y
N Quick Reply
G
H
=
a