Determine a list of eight positive integers (not necessarily distinct) such that summing seven of them in all eight possible ways generates only seven distinct results: 418, 420, 423, 424, 426, 428 and 429.
Let the smallest number be n. Then, the numbers are n, n+2, n+5, n+6, n+8, n+10, n+11, with one of them being repeated. Let the repeat be n+x.
The smallest number obviously leaves out n+11. Hence, adding the other 7,
7n+31+x = 418 = 7*59+5.
=> 31+x = 5mod7
=> x = 2 0r 9 => x =2
=> 7n+33 = 418 => n = 55.
So, the numbers are 55, 57, 57, 60, 61, 63, 65, 66.
Nirmal

Posted by nirmal
on 20060808 17:36:32 