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Sum Inverse LCMs (Posted on 2008-10-03) Difficulty: 3 of 5
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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re(2): a stab (proof) | Comment 5 of 7 |
(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.

  Posted by Steve Herman on 2008-10-04 01:06:44

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