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 Sum Inverse LCMs (Posted on 2008-10-03)
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

Note: [x, y] represents the LCM of x and y.

 No Solution Yet Submitted by K Sengupta Rating: 3.5000 (2 votes)

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 Extension (and ideas) | Comment 6 of 7 |
Only having 4 terms to deal with means that it's not too hard to just eliminate cases like the ones you suggested. I think it might be more interesting to deal with a sequence of any finite length

I think your case shows what happens in general it seems like 1/3 (or other fractions) are significantly lower than 1/2, and the rest of the sequence has to decay at a similar rate.

One way to show 1, 2, 4, 8, 16... is the best case is to compare it with the rest of the sequences possible. Call "P, Q, R, S, T, ..." the sequence, and [P, Q] -1 + [Q, R] -1 + [R, S] -1 + [S, T] -1 the goal equation.

It's clear that switching any number in the sequence to a higher number (and adjusting the following terms accordingly) will result in a lesser value of the goal equation, (similar to why the sequence must start with 1). So we need only consider switching a number to a lower term and adjusting accordingly.

Since 1/6 1/6 1/12 ... ends with 1/2-1/12 instead of 1/2-1/16, then it's going to be smaller. One would just need to prove this in the general case -- that since the last term here is smaller, the geometric series would have a last term smaller than whatever it would be in the goal case of 1+1/2+1/4+...

 Posted by Gamer on 2008-10-04 13:51:26

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