let a1, a2, . . . , a2023 be nonnegative real numbers such that a1 + a2 + . . . + a2023 = 100. let a = {(i, j) | 1 ≤ i ≤ j ≤ 2023, aiaj ≥ 1}. prove that |a| ≤ 5050 and determine when the equality holds.