Determine the total number of ways in which 111 can be written as a sum of three distinct nonzero integers which are in geometric sequence.

Prove that there are no others.

1 -11  121
1  10  100
27 -63  147
27  36  48
37 -74  148
48 -84  147
100 -110  121

The list is from the following program, but duplicates (such as in reverse order) have been removed manually.

FOR a = -5000 TO 5000
FOR b = a - 1 TO -5000 STEP -1
c = 111 - a - b
IF c <> b AND a <> b AND a <> c THEN
    IF a * c = b * b THEN PRINT a; b; c: ct = ct + 1
    IF a * b = c * c THEN PRINT a; c; b: ct = ct + 1
    IF b * c = a * a THEN PRINT b; a; c: ct = ct + 1
END IF
NEXT
NEXT

Posted by Charlie on 2013-08-07 12:45:22