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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
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]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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Peter   4
N Yesterday at 3:29 AM by sansgankrsngupta
Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) + 2\cos(2 \pi \alpha) = 0$. Prove that $ \alpha = \frac {2}{3}$.
4 replies
Peter
May 25, 2007
sansgankrsngupta
Yesterday at 3:29 AM
J
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R 3
Peter   13
N Mar 2, 2025 by EVS383
Prove no three lattice points in the plane form an equilateral triangle.
13 replies
Peter
May 25, 2007
EVS383
Mar 2, 2025
J
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A 22
Peter   6
N Sep 21, 2024 by pie854
Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.
6 replies
Peter
May 25, 2007
pie854
Sep 21, 2024
J
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A 30
Peter   10
N Sep 21, 2024 by pie854
Show that if $n \ge 6$ is composite, then $n$ divides $(n-1)!$.
10 replies
Peter
May 25, 2007
pie854
Sep 21, 2024
J
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A 21
Peter   11
N Sep 21, 2024 by pie854
Let n be a positive integer. Show that the product of $ n$ consecutive positive integers is divisible by $ n!$
11 replies
Peter
May 25, 2007
pie854
Sep 21, 2024
J
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A 20
Peter   11
N Sep 21, 2024 by pie854
Determine all positive integers $n$ for which there exists an integer $m$ such that $2^{n}-1$ divides $m^{2}+9$.
11 replies
Peter
May 25, 2007
pie854
Sep 21, 2024
J
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A 17
Peter   13
N Sep 19, 2024 by pie854
Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.
13 replies
Peter
May 25, 2007
pie854
Sep 19, 2024
J
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A 18
Peter   10
N Sep 19, 2024 by pie854
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.
10 replies
Peter
May 25, 2007
pie854
Sep 19, 2024
J
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A 15
Peter   10
N Sep 17, 2024 by pie854
Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.
10 replies
Peter
May 25, 2007
pie854
Sep 17, 2024
J
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A 14
Peter   9
N Sep 17, 2024 by pie854
Let $n$ be an integer with $n \ge 2$. Show that $n$ does not divide $2^{n}-1$.
9 replies
Peter
May 25, 2007
pie854
Sep 17, 2024
J
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I 15
Peter   2
N May 9, 2015 by Pythagorasauras
Find the total number of different integer values the function \[f(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\left\lfloor \frac{5x}{3}\right\rfloor+\lfloor 3x\rfloor+\lfloor 4x\rfloor\] takes for real numbers $x$ with $0 \leq x \leq 100$.
2 replies
Peter
May 25, 2007
Pythagorasauras
May 9, 2015
J
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floor function
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Peter
3615 posts
Peter
#1 h  May 25, 2007, 12:25 AM • 3 Y
Y by UpvoteFarm, Adventure10, Mango247
Find the total number of different integer values the function \[f(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\left\lfloor \frac{5x}{3}\right\rfloor+\lfloor 3x\rfloor+\lfloor 4x\rfloor\] takes for real numbers $x$ with $0 \leq x \leq 100$.
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Peter
3615 posts
Peter
#2 h  Aug 30, 2007, 10:11 AM • 3 Y
Y by Adventure10, UpvoteFarm, Mango247
http://www.kalva.demon.co.uk/apmo/asoln/asol932.html
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Pythagorasauras
13 posts
Pythagorasauras
#3 h  May 9, 2015, 6:09 PM • 2 Y
Y by UpvoteFarm, Adventure10
The link doesn't work...can you post the solution?
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