Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
All classes start 7:30pm ET/4:30pm PT unless otherwise noted.

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0 replies
jwelsh
Jul 1, 2025
0 replies
Trigonometry equation practice
ehz2701   6
N 41 minutes ago by vanstraelen
There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

Leaderboard

problem set 1a

problem set 2a

problem set 2b
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

[center] $\frac{1}{2} = \cos 36 - \sin 18$. [/center]

Many of these seem to be reducible to this. The half can be written as $\cos 60 = \sin 30$, and $\cos 36 = \sin 54$, $\sin 18 = \cos 72$. This is proven in solution 1a-1. We will refer to this as Trick 1.
6 replies
ehz2701
Yesterday at 8:48 AM
vanstraelen
41 minutes ago
Easy geometry problem
menseggerofgod   5
N 2 hours ago by ehz2701
ABC is a right triangle, right at B, in which the height BD is drawn. E is a point on side BC such that AE = EC = 8. If BD is 6 and DE = k , find k
5 replies
menseggerofgod
Today at 2:47 AM
ehz2701
2 hours ago
Geometry easy
AlexCenteno2007   4
N 3 hours ago by AlexCenteno2007
In triangle ABC, if angle B=120°, AB=5u and BC=15u. Draw the interior bisector BE. Calculate BE
4 replies
AlexCenteno2007
Friday at 10:55 PM
AlexCenteno2007
3 hours ago
a cute combinatorics (?) problem
pzzd   2
N 4 hours ago by mudkip42
here’s a cute little problem that can be solved with combinatorics, but is also related to some very common sequences in mathematics :3

say you have a $2$-inch-wide rectangle of some length $l$, and a bunch of $2$x$1$ dominos. how many different ways can you completely cover the rectangle with dominos? you can place the dominos horizontally or vertically - for example, for a $2$-by-$3$ rectangle, a valid arrangement of dominos is $1$ vertical domino on the left and $2$ horizontal dominos on the right.

hope you find this interesting!
2 replies
pzzd
6 hours ago
mudkip42
4 hours ago
No more topics!
Folklore
Osim_09   2
N May 17, 2025 by pigeon123
Let ABCD be a circumscribed quadrilateral, which is also cyclic. Let I be the incenter, O the circumcenter, and E the intersection point of the diagonals of the quadrilateral. Prove that the points O, I, and E are collinear.
2 replies
Osim_09
Jan 21, 2025
pigeon123
May 17, 2025
Folklore
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Osim_09
36 posts
#1
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Let ABCD be a circumscribed quadrilateral, which is also cyclic. Let I be the incenter, O the circumcenter, and E the intersection point of the diagonals of the quadrilateral. Prove that the points O, I, and E are collinear.
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Osim_09
36 posts
#3
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Can anyone try to solve this problem?
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pigeon123
6 posts
#4
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let A`,B`,C`,D` be the intersections of AI,BI,CI,DI with the circumcircle.First,let`s see that A`C`,and B`D` are diameters in the circumcircle.To prove this we see that <C`OAB+<A`OB=2(<C`CB+<A`AB)=<A+<C=180,hence A`C` is a diameter.Similarly we show that B`D` is a diameter.Now,let DA` cut CB` at Z.Then by Pascal in B`CAA`DB we get Z,E,I collinear.And since A`C` and B`D` intersect at O,by Pascal in B`CC`A`DD` we get Z,I,O collinear.Hence E,I,O are collinear.
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