China Girls' Mathematical Olympiad

Question

(1) Can one divide the set 1, 2, , 96 into 32 subsets, each containing three elements, and the sums of the three elements in each subset are all equal? (2) Can one divide the set 1, 2, , 99 into 33 subsets, each containing three elements, and the sums of the three elements in each subset are all equal? (Posed by Liu Shixiong)

Accepted Answer

(1) No. As 1+2++96=96(96+1)2=4897, and 32 48 97. (2) Yes. The sum of the three elements in each set is 1+2++9933 = 99 (99+1)33 2 = 150. We can divide 1, 2, 3, , 66 into 33 pairs, such that the sums of these pairs form an arithmetic sequence: 1 + 50, 3 + 49, , 33 + 34, 2 + 66, 4 + 65, , 32 + 51. Hence, the following decomposition satisfies the requirement: 1, 50, 99, 3, 49, 98, , 33, 34, 83, 2, 66, 82, 4, 65, 81, , 32, 51, 67.