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.
If you start by writing everything in terms of A and B you get
C = AB/(2AB)
D = A^2B/(2AB)^2
E = AB^2/(2AB)^2
I played with excel a bit but didn't discover much.
You can also use the last to solve for B, C and D in terms of A and E, but this requires the quadratic formula and leads to:
B = 2A(Eħ
√(AE))/(EA)
I haven't found C and D yet.
This actually shows some promise since it implies AE is a perfect square among other things.

Posted by Jer
on 20110405 02:00:21 