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Distance between products with sums

Source: 2022 China TST, Test 2, P6

March 29, 2022
algebracombinatorics

Problem Statement

Let m,nm,n be two positive integers with m≥n≥2022m \ge n \ge 2022. Let a1,a2,…,an,b1,b2,…,bna_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n be 2n2n real numbers. Prove that the numbers of ordered pairs (i,j) (1≤i,j≤n)(i,j) ~(1 \le i,j \le n) such that ∣ai+bj−ij∣≤m |a_i+b_j-ij| \le m does not exceed 3nmlog⁡n3n\sqrt{m \log n}.
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