Question

Are there any pairs of positive integers a,b so that (a+b)/2, sqrt(ab), 2ab/(a+b) and sqrt((a^2+b^2)/2) (AM, GM, HM, QM) are all integers?

Answer 1

Hi, and welcome to QuestionCove!

I think I can answer the question with an example: 4 is a positive integer. So, what if a=4 and b=4?
(4+4)/2 = 4, a positive integer
sqrt(4*4) = 4, a positive integer
2*4*4/(4+4) = 4, a positive integer
sqrt((4^2+4^2)/2)) = 4, a positive integer
In fact, it seems that in general, as long as "a" and "b" are the same positive integer, any of the calculations above work. You can try the calculations with any equal pair and see for yourself. The problem does not specify that "a" and "b" must be distinct, does it?
Now, if "a" and "b" were not allowed to be equal, it may be a different story. Because of the way the problem is worded, I think this solution works

Answer 2

Oops I meant to add a and b are different

Answer 3

I want to say that there aren't any pairs that fit the criteria, but I'm not sure how to prove it, and I wouldn't be completely confident in that answer without a proof... I'll try thinking about it and see if I come up with anything
I do have a clarifying question: when the problem mentions AM, GM, HM, and QM, do you know what those stand for?

Answer 4

arithmetic mean, geometric mean, harmonic mean, quadratic mean

Answer 5

so of course by scaling by a suitable constant it suffices to find a b such that sqrt(ab) and sqrt(2(a^2+b^2)) are integers. Try to find a solution :)

Answer 6

use a=kx^2 and b=ky^2 @minesweeper

Answer 7

this reduces to x^4+y^4=2z^2, which should be easy

Answer 8

bruh your bad i already did that and i couldnt find solutions i checked up to x,y=600

Answer 9

sobad you couldn't even solve an open problem i'm disappointed in you

Answer 10

SO bad alecks bad

Answer 11

oren so bad

Answer 12

now do the 55 mohs