ka March Highlights and 2025 AoPS Online Class Information
jlacosta0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.
Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!
Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.
Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/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.
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Introduction to Programming with Python
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The proposition is proven, when , i.e. .
First, i will show that :
Therefore, you need estimate 1: (proof by estimate 2: and calculation)
So, you have to show that
, what is true for all .
Now, you have to show that .
Therefore, you need estimate 3: (proof by calculation).
So, you have to show that
Now you need estimate 4: (again proof by calculation) and you have to show that
, what is obviously true for alle
Because of that, is true and hence, is true.
Y byUpvoteFarm, Adventure10, Mango247, and 1 other user
I think it is not enough to prove that . For example, but . Here is my proof:
Since is never a square, there is no integer in the interval . I will proof that .
The first inequality can be proven by squaring and rearranging terms a few times. It's very boring, so I won't write it here. The second inequality is equivalent to , which is obvious after squaring. Hence, proved.
I am unsure about the above proof; the lower bound is not trivial, and actually false for real in the interval , so I don't think it's easily proven 'by squaring and rearranging terms a few times'. I agree with the upper bound , which follows from the concavity of the square root function, and from this the inequality can be shown. To show that we actually have the reverse inequality as well, I want to make use of the fact that is never a square, from which we get that for all . It therefore suffices to show .
To prove this inequality, I claim that for all . Taking this latter claim for granted at the moment, we deduce , and we are done.
So it comes down to proving , to finish off the proof. We will show this by starting with the inequality and then repeatedly add or multiply by something positive, factorize both sides, or take square roots.
This post has been edited 2 times. Last edited by Woett, Mar 25, 2023, 11:25 AM