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Sum and Ratio Sum (Posted on 2011-01-16) Difficulty: 3 of 5
Determine all possible quadruplets (A, B, P, Q) of positive integers, with P ≤ Q, that satisfy this system of equations:

A + B = 21, and:

A/P2 + B/Q2 = 1

Prove that these are the only quadruplets that exist.

No Solution Yet Submitted by K Sengupta    
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Possible solution | Comment 2 of 4 |

Simply recasting the equation as:


gives solutions {a,p,q} at {5,3,6}{12,4,6}{37,6,24}{41,6,12}{48,6,9}{85,9,36}{101,9,18}...etc. of which only the first two have a less than 21, compliant with the stipulations of the problem.

  Posted by broll on 2011-01-17 02:08:11
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