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
Area of triangle inside regular hexagon
smartvong   1
N 13 minutes ago by tom-nowy
The diagram shows a regular hexagon $ABCDEF$. $P$ is a point on the line segment $CD$, such that the areas of $\triangle PAF$ and $\triangle PEF$ are $64$ and $48$ respectively. Find the area of $\triangle PAB$.

IMAGE
1 reply
smartvong
an hour ago
tom-nowy
13 minutes ago
J
H
Geometry Proof
strongstephen   14
N 37 minutes ago by strongstephen
Proof that choosing four distinct points at random has an equal probability of getting a convex quadrilateral vs a concave one.
not cohesive proof alert!

NOTE: By choosing four distinct points, that means no three points lie on the same line on the Gaussian Plane.
NOTE: The probability of each point getting chosen don’t need to be uniform (as long as it is symmetric about the origin), you just need a way to choose points in the infinite plane (such as a normal distribution)

Start by picking three of the four points. Next, graph the regions where the fourth point would make the quadrilateral convex or concave. In diagram 1 below, you can see the regions where the fourth point would be convex or concave. Of course, there is the centre region (the shaded triangle), but in an infinite plane, the probability the fourth point ends up in the finite region approaches 0.

Next, I want to prove to you the area of convex/concave, or rather, the probability a point ends up in each area, is the same. Referring to the second diagram, you can flip each concave region over the line perpendicular to the angle bisector of which the region is defined. (Just look at it and you'll get what it means.) Now, each concave region has an almost perfect 1:1 probability correspondence to another convex region. The only difference is the finite region (the triangle, shaded). Again, however, the actual significance (probability) of this approaches 0.

If I call each of the convex region's probability P(a), P(c), and P(e) and the concave ones P(b), P(d), P(f), assuming areas a and b are on opposite sides (same with c and d, e and f) you can get:
P(a) = P(b)
P(c) = P(d)
P(e) = P(f)

and P(a) + P(c) + P(e) = P(convex)
and P(b) + P(d) + P(f) = P(concave)

therefore:
P(convex) = P(concave)
14 replies
1 viewing
strongstephen
Tuesday at 4:54 AM
strongstephen
37 minutes ago
J
H
Inequalities
sqing   2
N 44 minutes ago by sqing
Let $ a,b>0 $ and $\frac{a}{a^2+3}+ \frac{b}{b^2+ 3} \geq \frac{1}{2} . $ Prove that
$$a^2+ab+b^2\geq 3$$$$a^2-ab+b^2 \geq 1 $$Let $ a,b>0 $ and $\frac{a}{a^3+3}+ \frac{b}{b^3+ 3}\geq \frac{1}{2} . $ Prove that
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
2 replies
sqing
Yesterday at 12:59 PM
sqing
44 minutes ago
J
H
Unknown triangle area
smartvong   0
an hour ago
The diagram shows a convex quadrilateral $ABCD$. The points $E$ and $F$ divide $AB$ into three equal parts while the points $G$ and $H$ divide $CD$ into three equal parts. The line segments $AH$ and $ED$ intersect at $I$. The line segments $CF$ and $BG$ intersect at $J$. Given that the areas of the triangles $AID$, $EHI$ and $FJG$ are $154$, $112$, and $99$ respectively, find the area of the triangle $BJC$.

IMAGE
0 replies
smartvong
an hour ago
0 replies
J
No more topics!
H
J
H
Pythagorean triples vs sine ratio?
Miranda2829   6
N Apr 10, 2025 by anticodon
I'm a bit confused about the

right angle 3 4 5 have a sine ratio of 0.6 and cosine of 0.8,

Do different lengths of right-angle triangles have different ratios?

how to get an actual angle of sine ?

thanks

6 replies
Miranda2829
Feb 27, 2025
anticodon
Apr 10, 2025
J
Pythagorean triples vs sine ratio?
G H J
Reveal topic tags
trigonometry
ratio
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.
Miranda2829
199 posts
Miranda2829
#1 h  Feb 27, 2025, 9:47 PM
Y by
I'm a bit confused about the

right angle 3 4 5 have a sine ratio of 0.6 and cosine of 0.8,

Do different lengths of right-angle triangles have different ratios?

how to get an actual angle of sine ?

thanks
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aBaoZi
14 posts
aBaoZi
#2 h  Feb 27, 2025, 11:29 PM
Y by
In a right triangle, the sine value of an angle is equal to the length of the side opposite of the angle divided by the length of the hypotenuse. The cosine value of an angle is the length of the side adjacent to the angle divided by the length of the hypotenuse.

Let’s say that this triangle with sides 3, 4, 5 has angles A, B, C; where A is the angle opposite of 3, B is the angle opposite of 4, and C is the right angle. The sine value for angle A would be the side opposite of angle A (which is 3) divided by the hypotenuse (which is 5), which is $3/5$, or $0.6$.

To get the actual angle value of angle A for this problem, you can type in sin^-1 (0.6) on the calculator.

Hope this helps!
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Miranda2829
199 posts
Miranda2829
#3 h  Feb 28, 2025, 2:12 AM
Y by
thanks, so what does the unit of circle's number do?

in what kind of question you need unit of circle? and what does it get for you ?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
char0221
140 posts
char0221
#4 h  Feb 28, 2025, 2:17 AM
Y by
A unit circle is required to find values of $\sin$ and $\cos$ that are greater than or even equal to $90^\circ$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
williamxiao
2514 posts
williamxiao
#5 h  Feb 28, 2025, 2:33 AM
Y by
Miranda2829 wrote:
thanks, so what does the unit of circle's number do?

in what kind of question you need unit of circle? and what does it get for you ?

The unit circle is useful for:
1. Telling you the exact cos/sin relations of angles that are multiples of 30 degrees or 45 degrees
2. Visualizing the relationship between (cos, sin) and (x,y) on a cartesian plane or an imaginary number on the complex plane
3. Relationships between angles and their rotations and reflections (like 150 degrees being the reflection of 30 degrees over the y axis)
This post has been edited 1 time. Last edited by williamxiao, Feb 28, 2025, 2:35 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Miranda2829
199 posts
Miranda2829
#6 h  Mar 1, 2025, 11:03 PM
Y by
thank u, how often we will see questions of sine angle not in unit of circle? like 38.67 or some other number which is not in unit of circle?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
anticodon
153 posts
anticodon
#7 h  Apr 10, 2025, 8:45 PM
Y by
on AMCs/contest problems not very often.
Usually just denote it as inverse sine or something and it should work
Z K Y
N Quick Reply
G
H
=
a