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.

(In reply to

re: a stab (proof) by Daniel)

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.