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Getting closer (Posted on 2012-11-25) Difficulty: 3 of 5
Find three distinct positive integers p, q and r such that the value of expression 1/p+1/q+1/r will be less than 1/2 but as close to it as possible.

Find four distinct positive integers p, q, r and t such that the value of expression 1/p+1/q+1/r+1/t will be less than 1 but as close to it as possible.

No Solution Yet Submitted by Ady TZIDON    
Rating: 4.5000 (2 votes)

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Solution solution | Comment 1 of 6

Start with 1/p+1/q+1/r=1, because we can see by inspection that, say, p and q are 2 and 3. That leaves 1/6, so the closest approach to 1 is 1/7 (making 1/6 just a bit smaller. 1/2+1/3+1/7=41/42, so p,q,r,t are 2,3,7,43 (just a bit smaller than 42). We can keep doing this, because these add up to 1805/1806, so the next fraction is 1/1807 and so on.

And the same reasoning applies to the first part: p=3,q=7,r=43.

It is (just) possible to coax Wolframalpha to compute the 9th and 10th members of the sequence without either running out of space on the input bar or exceeding the standard computation time. Any suggestions how to do it?

Edited on November 25, 2012, 11:58 am
  Posted by broll on 2012-11-25 11:57:38

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