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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:
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0 replies
jwelsh
Jul 1, 2025
0 replies
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Beautiful geo
justalonelyguy   0
3 minutes ago
Let \( \triangle ABC \) be an acute triangle with \( AB < AC \). The internal angle bisectors of angles \( \angle ABC \) and \( \angle ACB \) intersect at point \( I \). Let points \( K \) and \( L \) lie on segments \( IB \) and \( IC \), respectively, such that \( \angle KDL = 90^\circ \). Prove that \[ \angle KAL = \frac{1}{2} \angle BAC. \]
0 replies
justalonelyguy
3 minutes ago
0 replies
J
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Trigonometry equation practice
ehz2701   29
N 31 minutes ago by ehz2701
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 and Solved Problems

problem set 1a (1-10)

problem set 2a (1-20)

problem set 2b (1-20)
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.
29 replies
ehz2701
Jul 12, 2025
ehz2701
31 minutes ago
J
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[PMO19 Qualis] II.4 Minimum of rational function
aops-g5-gethsemanea2   1
N an hour ago by P0tat0b0y
If $b>1$, find the minimum value of $\frac{9b^2-18b+13}{b-1}$.

$\text{(a) }0\qquad\text{(b) }9\qquad\text{(c) }12\qquad\text{(d) }36$

Answer confirmation
1 reply
aops-g5-gethsemanea2
an hour ago
P0tat0b0y
an hour ago
J
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[PMO19 Qualis] II.6 100th digit
aops-g5-gethsemanea2   0
an hour ago
What is the 100th digit of the following sequence?
$$1 ~4~ 9 ~16~ 25 ~36~ 49 ~64~ 81 ~100~ \dots$$
$\text{(a) 6\qquad(b) 7\qquad(c) 8\qquad(d) 9}$

Answer confirmation
0 replies
aops-g5-gethsemanea2
an hour ago
0 replies
J
No more topics!
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Domain of Nested Functions
pensive   1
N Jun 22, 2025 by pensive
Given $f(x) = \log_2 (x) - 1$ and $g(x) = -x^2+x+2$, for what integer $x$ is $f(g(f(x)))$ positive?
Answer
1 reply
pensive
Jun 22, 2025
pensive
Jun 22, 2025
J
Domain of Nested Functions
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#1 h  Jun 22, 2025, 3:09 PM
Y by
Given $f(x) = \log_2 (x) - 1$ and $g(x) = -x^2+x+2$, for what integer $x$ is $f(g(f(x)))$ positive?
Answer
This post has been edited 1 time. Last edited by pensive, Jun 22, 2025, 3:10 PM
Reason: Misleading text
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#2 h  Jun 22, 2025, 3:35 PM
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