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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
J
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Colouring numbers
kitun   2
N a minute ago by quasar_lord
What is the least number required to colour the integers $1, 2,.....,2^{n}-1$ such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers $i, j$ such that $1<=i<=j<=2^{n}-1$, there will be a colour coloring exactly one integer from the set $i, i+1,.... , j-1, j$.
2 replies
kitun
Nov 15, 2021
quasar_lord
a minute ago
J
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Mathhhhh
mathbetter   4
N 3 minutes ago by Amkan2022
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
4 replies
mathbetter
Yesterday at 11:21 AM
Amkan2022
3 minutes ago
J
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Finally hard NT on UKR MO from NT master
mshtand1   3
N 17 minutes ago by Jupiterballs
Source: Ukrainian Mathematical Olympiad 2025. Day 1, Problem 11.4
A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)?

Proposed by Oleksii Masalitin
3 replies
mshtand1
Mar 13, 2025
Jupiterballs
17 minutes ago
J
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Inspired by my own results
sqing   2
N 26 minutes ago by lbh_qys
Source: Own
Let $ a,b,c\geq 2.$ Prove that
$$ (a+1)(b+2)(c +1)-3 abc\leq 12$$$$ (a+1)(b+2)(c +1)-\frac{7}{2}abc\leq 8$$$$ (a+1)(b+3)(c +1)-\frac{15}{4}abc\leq 15$$$$ (a+1)(b+3)(c +1)-4abc\leq 13$$
2 replies
sqing
2 hours ago
lbh_qys
26 minutes ago
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No more topics!
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J
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3a^2b+16ab^2 is perfect square for primes a,b >0
parmenides51   5
N Wednesday at 6:31 PM by ali123456
Source: 2020 Greek JBMO TST p3
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
5 replies
parmenides51
Nov 14, 2020
ali123456
Wednesday at 6:31 PM
J
3a^2b+16ab^2 is perfect square for primes a,b >0
G H J
Reveal topic tags
number theory
Perfect Square
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G H BBookmark kLocked kLocked NReply
Source: 2020 Greek JBMO TST p3
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parmenides51
30628 posts
parmenides51
#1 h  Nov 14, 2020, 8:12 PM • 3 Y
Y by Mango247, Mango247, Mango247
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
This post has been edited 1 time. Last edited by parmenides51, Nov 14, 2020, 8:13 PM
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i3435
1349 posts
i3435
#2 h  Nov 14, 2020, 8:22 PM
Y by
Sol
This post has been edited 1 time. Last edited by i3435, Nov 14, 2020, 8:30 PM
Reason: i missed a solution
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pi_quadrat_sechstel
582 posts
pi_quadrat_sechstel
#3 h  Nov 14, 2020, 8:22 PM
Y by
$ab(3a+16b)$ is a perfect sqare. If $a=b$, the number $19a$ is a perfect square. Since $a$ is prime $a=b=19$. If $a\neq b$, we must have $a\mid 3a+16b$ and $b\mid 3a+16b$. Since $a,b$ are different primes, we have $a\mid16$ and $b\mid3$. Thus $a=2,b=3$. Obviously, $(2,3),(19,19)$ are solutions.
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parmenides51
30628 posts
parmenides51
#4 h  Nov 14, 2020, 8:25 PM
Y by
i3435 wrote:
What's the structure of the greek tst test? This seems offly trivial for a $3$.
this is JBMO TST problem, for the Junior Balkan MO (JBMO) ,
it comes right after the Junior National Math Olympiad (it has 3 rounds, the 3rd is the final)
This post has been edited 3 times. Last edited by parmenides51, Nov 14, 2020, 8:26 PM
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Euler...
10 posts
Euler...
#5 h  Mar 8, 2024, 8:28 PM
Y by
$A=ab(3a+16b)=k^2$ there are 2 cases , if a=b then $(a;b)=(19;19)$ , second case is $a\neq b$ GCD(a;b)=1 a|3a+16 also b|3a+16b and here there is only one solution (2;3)=(a;b)
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ali123456
44 posts
ali123456
#6 h  Wednesday at 6:31 PM
Y by
My solution
This post has been edited 1 time. Last edited by ali123456, Wednesday at 6:34 PM
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