Each of P, Q, R, S and T
are positive integers
with P < Q < R < S < T
. Determine the maximum value
of the following expression.
[P, Q] -1 + [Q, R] -1 + [R, S] -1 + [S, T] -1
: [x, y] represents the LCM
of x and y.
(In reply to re: a stab (proof)
I agree that the maximum value is 15/16, based on (1,2,4,8,16), which leads to a value of 1/2 + 1/4 + 1/8 + 1/16 = 15/16
However, I don't think that Daniel has proved it. The problem is right at the beginning of the proposed proof, where it is concluded that R is a multiple of Q. This needs to be proved more rigorously.
Clearly P has to be 1. But what if R is less than 2Q?
Consider, for instance, (1,2,3,6,12). This leads to the value of 1/2 + 1/6 + 1/6 + 1/12 = 11/12. Note that the last two terms, 1/6 + 1/12, are greater than 1/8 + 1/16. The total value of 11/12 is less than 15/16, but only because the larger 3rd and 4th terms do not compensate for the smaller 2nd term.
Are there some values which will lead to a bigger value, than 5/16, without R being a multiple of Q? This leads to a smaller second term, but couldn't later terms compensate for this deficiency? The answer is no, but it hasn't been proved yet.