ka May Highlights and 2025 AoPS Online Class Information
jlacosta0
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]
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Introduction to Programming with Python
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Initially, my problem was : is it true that if we fix positive such that , then there exists positive such that converges and diverges ?
Thinking about the continuous case : if is continuous, positive with , does continuous and positive exist on such that converges and diverges ?
To the last question, the answer seems to be yes if is in the class, increasing : I chose . With this idea, I had the idea to define but it is not clear that it is ok, even if is increasing.
Now I have some questions !
1) The main problem : is it true that if we fix positive such that , then there exists positive such that converges and diverges ? And if is increasing ?
2) is it true that if we fix positive increasing such that
and , then diverges ?
3) is it true that if we fix positive increasing such that
and , then converges ?
4) if is positive increasing and such that
and does not converge to , can diverge ?
5) for the continuous case, is it true if we suppose only to be continuous ?
is similar, over , to , where .
Then we may assume that is in Jordan form. Here, we consider only the case .
Then is a block-matrix where the entries are matrices in the form .
Let ; since the pairwise commute, , where is obtained by developing the determinant by considering the as elements.
Finally , where is a polynomial of degree in .
The general case -the sequel of post #2-
We may assume (see above) that and .
Then and , where .
Note that is the matrix in post #2 and that is a block-matrix where the entries are matrices in the form (then nilpotent).
As in post #2, but here does not depend on
the matrix , that is, .
PS. If , then , where .
Then and, by continuity in ,.
This post has been edited 1 time. Last edited by loup blanc, Apr 17, 2025, 8:59 PM