SAUDI ARABIAN IMO Booklet 2023

Question

Determine whether or not it is possible to partition the set of positive integers in infinite subsets

A_{1}, A_{2}, \dots

such that for every positive integer

k

, the sum of elements of

A_{k}

is

k + 2023

.

Remark: a partition of a set

X

is a collection of subsets of

X

such that every element of

X

is contained in exactly one the subsets.

Accepted Answer

The answer is No. Suppose such partition exists. Then for every positive integer k, we have B_k = A_1 A_2 A_k 1, 2, , k + 2023, since all elements of A_i are at most i + 2023 for every i 1, 2, , k and _b B_k b = _i=2024^k+2023 i < _i=1^k+2023 i. Now let t_k be the minimum positive integer that not in B_k. Then we have _i=2024^k+2023 i = _b B_k b ( _i=1^k+2023 i ) - t_k, which implies that t_k _i=1^2023 i, for every integer k. But that cannot happen if A_1, A_2, is a partition of the positive integers.