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NT, Combo, or Geo?

Source: 2020 USOMO Problem 4, USOJMO Problem 5

June 21, 2020
USOMOusojmoUSO(J)MO2020 USAMO2020 USAJMO

Problem Statement

Suppose that (a1,b1),(a_1,b_1), (a2,b2),(a_2,b_2), …,\dots, (a100,b100)(a_{100},b_{100}) are distinct ordered pairs of nonnegative integers. Let NN denote the number of pairs of integers (i,j)(i,j) satisfying 1≤i<j≤1001\leq i<j\leq 100 and ∣aibj−ajbi∣=1|a_ib_j-a_jb_i|=1. Determine the largest possible value of NN over all possible choices of the 100100 ordered pairs.
Proposed by Ankan Bhattacharya
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