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
Final problem
Cats_on_a_computer   14
N a few seconds ago by HappyOreoGuineaPig24
Source: Own
This is likely the last post I will make in my life.

Consider an 18 year old who has no purpose, talents, or friends; a living waste of space, an unsightly chthonic maggot with less of a right to live than a grasshopper. Note that this person is so desperate, he writes his suicide note on a math forum of all places, because nobody around him would bother reading one. We define a *solution* to this individual’s woes as a termination. What is the optimal play by this individual to reach a solution with the least amount of pain?

Solution (sketch): we construct a 1 dimensional CW-complex consisting of a single circle $S_1$, and an interval glued with one of its endpoints to the circle.

See you later, space cowboy…
14 replies
Cats_on_a_computer
36 minutes ago
HappyOreoGuineaPig24
a few seconds ago
An interesting combination problem
Math291   4
N a few seconds ago by GA34-261
Given a unit square grid of size 4×6 as shown in the figure below, an ant crawls from point A. Each time it moves, it crawls along the side of a unit square to an adjacent grid point.
IMAGE
How many number of ways to complete a path so that after exactly 12 moves, it stops at position B?
4 replies
+1 w
Math291
15 minutes ago
GA34-261
a few seconds ago
Finite projective plane
EthanWYX2009   0
3 minutes ago
Source: 2023 December 谜之竞赛-3
A positive integer is called Christmas number if it can be expressed as the sum of squares of pairwise differences of $1893$ integers. Determine the minimum positive integer \( a \) that is not a perfect square, such that any multiple \( an \) of a Christmas number \( n \) remains a Christmas number.

Proposed by Xiuyi Chen from Shanghai High School
0 replies
EthanWYX2009
3 minutes ago
0 replies
inequality
SunnyEvan   3
N 3 minutes ago by SunnyEvan
Source: Own
Let $ x \in (\frac{\pi}{2}-1, 1) $, try to prove or disprove that :
$$ \frac{(\sqrt2 cosx -1)^2}{cos2x+tan\frac{\pi}{8}}-\frac{(\sqrt2 sinx -1)^2}{cos2x-tan\frac{\pi}{8}} \geq \frac{1}{2}(\frac{tanx-1}{tanx+1})^2 $$
3 replies
SunnyEvan
Yesterday at 1:24 PM
SunnyEvan
3 minutes ago
No more topics!
construct triangle
rogue   1
N Sep 10, 2008 by mr.danh
Source: Ukrainian journal contest, problem 329, by Grygoriy Filippovskyy
Construct triangle $ ABC$ given points $ O_A$ and $ O_B,$ which are symmetric to its circumcenter $ O$ with respect to $ BC$ and $ AC,$ and the straight line $ h_A,$ which contains its altitude to $ BC.$
1 reply
rogue
Sep 9, 2008
mr.danh
Sep 10, 2008
construct triangle
G H J
Source: Ukrainian journal contest, problem 329, by Grygoriy Filippovskyy
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rogue
554 posts
#1 • 2 Y
Y by Adventure10, Mango247
Construct triangle $ ABC$ given points $ O_A$ and $ O_B,$ which are symmetric to its circumcenter $ O$ with respect to $ BC$ and $ AC,$ and the straight line $ h_A,$ which contains its altitude to $ BC.$
Z K Y
The post below has been deleted. Click to close.
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mr.danh
635 posts
#2 • 2 Y
Y by Adventure10, Mango247
rogue wrote:
Construct triangle $ ABC$ given points $ O_A$ and $ O_B,$ which are symmetric to its circumcenter $ O$ with respect to $ BC$ and $ AC,$ and the straight line $ h_A,$ which contains its altitude to $ BC.$
The perpendicular bisector of $ O_AO_B$ meets the line $ h_A$ at the orthocenter H. The circle through H centered $ O_A$ meets the circle through H centered $ O_B$ at C. The circle $ (O_A)$ meets the perp line from C to $ h_A$ at B.
Z K Y
N Quick Reply
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