Each of P, Q, R, S and (P+S), with P < Q < R < S, is a non leading zero 10digit 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 (09) 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^2Q A*ZR (A^2+Z^2)/2S Z^2P+S (A^2+Z^2)*2The 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 20090723 12:17:56 