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 arithmetic sequence, and:
(ii)
B, C and D (in his order) are in geometric sequence, and:
(iii)
C, D and E (in this order) are in
harmonic sequence.
Determine the
minimum value of
(EA) such that there are
precisely two quintuplets (A, B, C, D, E) that satisfy all the given conditions.
Note: Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.
(In reply to
computer solutions  no proof by Charlie)
Using substitution I was able to derive the equation,
[EA] = (4/A)*x^{2} + 4*x,
such that x is the common difference between A and B.
With this bit of information, and limiting my search to where
[EA] = 64, I was able to search beyond A=200,000 to find if there were any subsequent sets. Yet, due to limitations, my search limited A to not much greater than 2,000,000, i.e., only a 10fold increase in range. No values for (A, x) other than (8, 8) and (36, 12) such that D also was an integer, i.e.,
x^{2} modulo (A + x) = 0, was found.

Posted by Dej Mar
on 20081029 10:09:47 