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 May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/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
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
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, 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
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
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
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
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

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
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
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
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
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
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

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

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
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
J
H
find angle
TBazar   2
N 5 minutes ago by sunken rock
Given $ABC$ triangle with $AC>BC$. We take $M$, $N$ point on AC, AB respectively such that $AM=BC$, $CM=BN$. $BM$, $AN$ lines intersect at point $K$. If $2\angle AKM=\angle ACB$, find $\angle ACB$
2 replies
+1 w
TBazar
2 hours ago
sunken rock
5 minutes ago
J
H
Geometry marathon
HoRI_DA_GRe8   845
N 14 minutes ago by leon.tyumen
Ok so there's been no geo marathon here for more than 2 years,so lets start one,rules remain same.
1st problem.
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR=90°$ and let $H$ and $K$ be the feet of altitudes from $Q$ to the lines $PR$ and $PS$,.Prove $HK$ bisects $QS$.
P.s._eeezy ,try without ss line.
845 replies
HoRI_DA_GRe8
Sep 5, 2021
leon.tyumen
14 minutes ago
J
H
My Unsolved Problem
ZeltaQN2008   0
16 minutes ago
Let \(f:[0,+\infty)\to\mathbb{R}\) be a function which is differentiable on \([0,+\infty)\) and satisfies
\[ \lim_{x\to+\infty}\bigl(f'(x)/e^x\bigr)=0. \]Prove that
\[ \lim_{x\to+\infty}\bigl(f(x)/e^x\bigr)0. \]
0 replies
ZeltaQN2008
16 minutes ago
0 replies
J
H
polonomials
Ducksohappi   0
29 minutes ago
$P\in \mathbb{R}[x] $ with even-degree
Prove that there is a non-negative integer k such that
$Q_k(x)=P(x)+P(x+1)+...+P(x+k)$
has no real root
0 replies
Ducksohappi
29 minutes ago
0 replies
J
H
[PMO24 Qualifying II.3] Positive Divisors Problem
kae_3   3
N Feb 18, 2025 by MineCuber
Let $m$ and $n$ be relatively prime positive integers. If $m^3n^5$ has $209$ positive divisors, then how many positive divisors has $m^5n^3$ have?

Answer Confirmation
3 replies
kae_3
Feb 16, 2025
MineCuber
Feb 18, 2025
J
H
Sequence of Numbers
4everwise   6
N Feb 4, 2025 by megarnie
A sequence of numbers $x_{1},x_{2},x_{3},\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50}=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
6 replies
4everwise
Jan 5, 2007
megarnie
Feb 4, 2025
J
H
asdf1434 Mock AIME #9
P_Groudon   6
N Jan 29, 2025 by Squ_red
In regular tetrahedron $ABCD$ of side length 1, let $P$ be a point on line $AB$ and let $Q$ be a point on the line through $C$ and the midpoint of $AD$. The least possible value of $PQ^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
6 replies
P_Groudon
May 8, 2024
Squ_red
Jan 29, 2025
J
H
asdf1434 Mock AIME #6
P_Groudon   8
N Jan 20, 2025 by clarkculus
Let $ABCD$ be a trapezoid with $AB \parallel CD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively, and let $E$ be the intersection of diagonals $AC$ and $BD$. If $EM = 1$, $EN = 3$, $CD = 10$, and $\frac{AD}{BC} = \frac{3}{4}$, then $AD^2 + BC^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
8 replies
P_Groudon
May 8, 2024
clarkculus
Jan 20, 2025
J
H
jungle diff ff 15 feed
hexuhdecimal   3
N Jan 17, 2025 by sdfgfjh
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute \[\sum_{i=1}^{\phi(2023)} \frac{gcd(i, \phi(2023))}{\phi(2023)}\].
3 replies
hexuhdecimal
Jan 17, 2025
sdfgfjh
Jan 17, 2025
J
H
Fractions
Jia_Le_Kong   3
N Jan 12, 2025 by Lankou
I write down a list of all fractions $\frac{m}{n}$, where $m, n$ are relatively prime positive integers and $n$ is at most $2025$.

The list is sorted in ascending order. What is the fraction that comes right after $\frac{26}{49}?$
3 replies
Jia_Le_Kong
Jan 12, 2025
Lankou
Jan 12, 2025
J
H
Exercise 4. Find all distinct positive integers a, b, c that have relatively pri
hoangvu1009   3
N Dec 16, 2024 by AbhayAttarde01
Exercise 4. Find all distinct positive integers a, b, c that have relatively prime factors such that the sum of any two numbers is always divisible by the third number.
3 replies
hoangvu1009
Dec 15, 2024
AbhayAttarde01
Dec 16, 2024
J
H
Infinate Geometric Series
4everwise   6
N Dec 13, 2024 by pateywatey
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.
6 replies
4everwise
Nov 13, 2005
pateywatey
Dec 13, 2024
J
H
Algebra?
SomeonecoolLovesMaths   1
N Nov 24, 2024 by alexheinis
Let $a,b,c$ be positive reals such that $abc + a + b = c$ and $$\frac{19}{\sqrt{a^2+1}} + \frac{20}{\sqrt{b^2+1}} = 31.$$The maximum possible value of $c^2$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
1 reply
SomeonecoolLovesMaths
Nov 22, 2024
alexheinis
Nov 24, 2024
J
H
Geometry
SomeonecoolLovesMaths   1
N Nov 22, 2024 by sdfgfjh
Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, $CA = 7$. Let $O$ be the circumcenter of $\bigtriangleup ABC$ and let $P$ be a point such that $\overline{AB} \perp \overline{BP}$ and $\overline{AC} \perp \overline{AP}$. If lines $OP$ and $BC$ intersect at $T$, then the length $BT$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1 reply
SomeonecoolLovesMaths
Nov 22, 2024
sdfgfjh
Nov 22, 2024
J
H
J
H
general form
pennypc123456789   2
N Apr 20, 2025 by sqing
If $a,b,c$ are positive real numbers, $k \ge 3$ then
$$ \frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$
2 replies
pennypc123456789
Apr 18, 2025
sqing
Apr 20, 2025
J
general form
G H J
Reveal topic tags
inequalities
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.
pennypc123456789
38 posts
pennypc123456789
#1 h  Apr 18, 2025, 6:18 AM
Y by
If $a,b,c$ are positive real numbers, $k \ge 3$ then
$$ \frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42045 posts
sqing
#2 h  Apr 18, 2025, 9:10 AM
Y by
Let $ a,b,c>0 . $ Prove that
$$\frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$Where $k\geq 2.5826 .$
*
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sqing
42045 posts
sqing
#3 h  Apr 20, 2025, 1:43 PM
Y by
SunnyEvan wrote:
Let $ f(a,b,c)=\frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} $
try to expand and use CD3 theorem : $ f(1,1,1) \geq \dfrac{6}{k+2} $ and $ f(a,b,0) \geq 0 $
equality case :$ (a,b,c)=(t,t,t) $ where $t>0$
https://artofproblemsolving.com/community/c6h3552321p34603474
Z K Y
N Quick Reply
G
H
=
a