(a) Can the sum of 12345 odd pos-
itive integers be 12345678 ? Prove or disprove.
(b) Prove that for every possible re-arrangement (b1, b2, . . . ,b999) of the
numbers (1, 2, . . . , 999), the product
(b1 ? 1)(b2 ? 2)(b3 ? 3) · . . . · (b999 ? 999) is even.
