Five positive integers A, B, C, D and E, with A < B < C < D < E, are such that:

(i) A, B and C (in this order) are in harmonic sequence, and:

(ii) B, C and D (in this order) are in geometric sequence, and:

(iii) C, D and E (in this order) are in arithmetic sequence.

Determine the minimum value of (E-A) such that there are precisely three quintuplets (A, B, C, D, E) that satisfy all the given conditions.

Let X=E-A

Then X=(4A^2b)/(A-b)^2 (with b being the difference between the reciprocals of A and B) 'works' for {A,b} = {32,16}{144,36}{392,49} with (E-A)=256.

Posted by broll on 2011-04-06 10:28:24