Each of P, Q, R, S and (P+S), with P < Q < R < S, is a non leading zero 10-digit base ten positive integer containing each of the digits from 0 to 9 exactly once. It is known that R is the arithmetic mean of P and S, and Q is the geometric mean of P and S.

Determine the minimum value of P and the maximum value of S.

The following comment was made after viewing data derived from a computer program which is noted in a following comment.  That data very conveniently fitted into the paramaters for P, Q, R and S.

While convenient this is flawed:
     eg, If P = 2 and S = 32 then Q= √64 = 8.
--------------------------------------------------------------------

Comments made from viewing data:

There are 86 pandigitals (0-9) which are squares.

P is one of these at the lower end while S is also one at the higher end.

The range is from 1026753849 (32043*32043) to 9814072356 (99066*99066).

S cannot be that latter value as P+S would become a 10^10 number.

Since P and S are squares Q is the product of their square roots (GM).

Let A be the square root of P and Z be the square root of  S.

The conditions to be met are then:

P           A^2
Q          A*Z
R          (A^2+Z^2)/2
S          Z^2
P+S       (A^2+Z^2)*2

The values and range were not calculated by me but gleaned from a site which offered a program which ran through all PD's looking for squares!  Did it in a few seconds - think the program was written in Pascal; I'll add that acknowledgement later.

Seems to me that given the range of squares, and that they are all 5 digits, a program using a low powered language has a chance of offering a result.

Edited on July 23, 2009, 10:48 pm

Edited on July 23, 2009, 10:50 pm

Posted by brianjn on 2009-07-23 12:17:56