Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
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
Interesting inequalities
sqing   1
N 12 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , ab+bc+kca =k+2.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq  \frac{k^2+k+2+2k\sqrt{k+2}}{(k+1)^2}$$Where $ k\in N^+.$
Let $ a,b,c\geq 0 , ab+bc+ca =3.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq 1+\frac{\sqrt 3}{2}$$Let $ a,b,c\geq 0 , ab+bc+2ca =4.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq  \frac{16}{9}$$
1 reply
1 viewing
sqing
14 minutes ago
sqing
12 minutes ago
Non-homogeneous degree 3 inequality
Lukaluce   3
N 12 minutes ago by Sh309had
Source: 2024 Junior Macedonian Mathematical Olympiad P1
Let $a, b$, and $c$ be positive real numbers. Prove that
\[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\]When does equality hold?

Proposed by Petar Filipovski
3 replies
Lukaluce
Apr 14, 2025
Sh309had
12 minutes ago
Distinct Integers with Divisibility Condition
tastymath75025   17
N 19 minutes ago by quantam13
Source: 2017 ELMO Shortlist N3
For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$.

Proposed by Daniel Liu
17 replies
tastymath75025
Jul 3, 2017
quantam13
19 minutes ago
IMO 2011 Problem 1
Amir Hossein   102
N 19 minutes ago by Jupiterballs
Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq  i < j \leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

Proposed by Fernando Campos, Mexico
102 replies
Amir Hossein
Jul 18, 2011
Jupiterballs
19 minutes ago
No more topics!
altitude, median, bisector, and trisector
Raja Oktovin   8
N Jul 15, 2014 by nawaites
Source: Indonesia IMO 2010 TST, Stage 1, Test 1, Problem 4
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP=\angle PBQ=\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b) calculate $ \dfrac{BP}{CH}$.
Soewono, Bandung
8 replies
Raja Oktovin
Nov 12, 2009
nawaites
Jul 15, 2014
altitude, median, bisector, and trisector
G H J
Source: Indonesia IMO 2010 TST, Stage 1, Test 1, Problem 4
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Raja Oktovin
277 posts
#1 • 1 Y
Y by Adventure10
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP=\angle PBQ=\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b) calculate $ \dfrac{BP}{CH}$.
Soewono, Bandung
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Raja Oktovin
277 posts
#2 • 2 Y
Y by Adventure10, Mango247
it is just Click to reveal hidden textto tackle this problem.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vo Duc Dien
341 posts
#3 • 1 Y
Y by Adventure10
I don't know about this, but I have \[ \angle ABP=\angle PBQ=\angle QBC, \] = 8.948922449° (yes, that ugly angle) and $ \frac{BP}{CH} $ = 2.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vo Duc Dien
341 posts
#4 • 2 Y
Y by Adventure10, Mango247
Yes, the angle measurement of 8.948922449.....° is correct.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Vo Duc Dien
341 posts
#5 • 2 Y
Y by Adventure10, Mango247
Hint: When the sum of two angles equals 90 degrees, the sum of the squares of the sine values of the angles equals 1.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
littletush
761 posts
#6 • 1 Y
Y by Adventure10
by a simple trigonometric calculation we get
$\angle A=63.15323236...,\angle B=26.84676764...$
whose sum is 90.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hatchguy
555 posts
#7 • 2 Y
Y by Adventure10, Mango247
I don't know how valid is this argument but its probably the same as trig ceva.

From $\angle PBA = \angle QBC$ and $\angle PAC = \angle QAB$ we have that $P$ and $Q$ are isogonal conjugates. Therefore , $CM$ is the reflection of $CH$ over the bisector of $\angle ACB$ and therefore we get that the circumcenter of $ABC$ lies in $CM$. Therefore we must have $\angle ACB = 90$. This concludes part (a).

Now let $\angle CAB = \alpha$. By easy angle chase we get $\angle ABP = 30 - \frac{\alpha}{3}$, $\angle PBC = \angle 60 - \frac{3\alpha}{2}$.

We have $BP = \frac {BP}{\sin (60 + \frac{\alpha}{3})} =\frac{BC \cdot \sin \alpha}{\sin (60 + \frac{\alpha}{3})}$ and $CH = BC \cdot \cos \alpha$ and therefore we obtain \[\frac{BP}{CH} = \frac{\sin \alpha}{\cos \alpha  \cdot \sin 60 + \frac{\alpha}{3}} (*)\]


From trig ceva for cevians $AP,CP,BP$ we get $\sin \alpha = \frac{ \cos \alpha \cdot \sin 60 - \frac{2\alpha}{3}}{\sin 30 - \frac{\alpha}{3}}$ so plugging this in $(*)$ gives us

\[ \frac{BP}{CH} = \frac{\sin 60 - \frac{2\alpha}{3}}{ \sin 30 - \frac{\alpha}{3} \cdot \sin 60 + \frac{\alpha}{3}} = \frac{\sin 60 - \frac{2\alpha}{3}}{ \sin 30 - \frac{\alpha}{3} \cdot \cos 30 - \frac{\alpha}{3}} = 2 \]
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
chronondecay
145 posts
#8 • 2 Y
Y by Adventure10, Mango247
Here's a synthetic solution to (b).

Reflect $P$ about $AB$ to $P'$. Then $\angle CBP = 2 \angle PBH = \angle PBP'$, so by similar triangles and two applications of Angle Bisector Theorem we have \[\frac{CB}{CH} = \frac{AC}{AH} = \frac{CP}{PH} = 2\frac{CP}{PP'} = 2\frac{CB}{P'B} = 2\frac{CB}{PB},\]thus $\frac{BP}{CH} = 2$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
nawaites
204 posts
#9 • 2 Y
Y by Adventure10, Mango247
Anyone can proof a?
Since i dont know how to use isogonal conjaquates
Z K Y
N Quick Reply
G
H
=
a