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Get one half by squaring (Posted on 2004-09-20) Difficulty: 4 of 5
Can you solve the following equation?

= 1/x + 1/y +...+ 1/z

All variables must be different, positive integers, and there must be a finite number of terms.

See The Solution Submitted by Federico Kereki    
Rating: 4.2500 (12 votes)

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Continue Beating the Horse... | Comment 26 of 31 |

I posed this problem to a bright freshman I met on campus, who found the following infinite solution in less than a minute:

Take all powers of 2, 3, and 5.

Note that the infinite geometric sequence 1/a^2, 1/a^4, 1/a^6, ... has sum 1/(a^2-1).

Thus the 2's have sum 1/3, the 3's have sum 1/8, and the 5's have sum 1/24.  The total sum is 1/3+1/8+1/24=1/2.

I was very impressed.

  Posted by David Shin on 2004-09-22 21:05:41
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