Problem 190

[Zkid]

Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers with x1 + x2 + ... + xm = m for which Pm = x1 * x2^2 * ... * xm^m is maximised.

For example, it can be verified that [P10] = 4112 ([ ] is the integer part function).

I must be misunderstanding this as it seems to be that, for example, S10 = (2, 2, 2, 2, 2) gives P10 = 32768 which is much greater. What am I misreading?

[jaap]

Let Sm = (x1, x2, ... , xm) be the m-tuple of positive real numbers with x1 + x2 + ... + xm = m for which Pm = x1 * x2^2 * ... * xm^m is maximised.

For example, it can be verified that [P10] = 4112 ([ ] is the integer part function).

I must be misunderstanding this as it seems to be that, for example, S10 = (2, 2, 2, 2, 2) gives P10 = 32768 which is much greater. What am I misreading?

S10 needs to have 10 numbers in it, x1...x10 that sum to 10. You are only using five.

[Zkid]

That makes a lot more sense - I read that pretty carelessly. Thanks

[PropagandaPanda]

From what I understand right now, Pm should be an integer since we need to compute its integer partition.

However, it seems that Pm is not an integer. Taking m=2, the simple corresponding function is:
y = x*(2-x)^2

which is maximized for x>0 and (2-x)>0 at (2/3, 32/27 ). 32/27 is not an integer. I must be misunderstanding something.

Thanks.

[TheEvil]

The point is, that P_m should not be integer. If your answer (a=32/27) is correct, then you have to take the greatest integer which is less or equal to a, in this case, you should type 1 as a correct answer.

[PropagandaPanda]

Ah I see. Thank you!

[PropagandaPanda]

For some reason, I had interpreted that as the Integer Partition function.

[philiplu]

There's a 19 Nov 2013 post by dcterr with 8 kudos but no permanent lock on it in the PE190 forum, and it's only a few posts from being deleted. Can it be made permanent? It's a really good explanation of the theory behind the problem which I refer to occasionally, and I'd hate to see it disappear.

https://projecteuler.net/thread=190;page=5#142817

[RobertStanforth]

Thanks - I've made it permanent.