MathDB
NT with congruences

Source: Russian TST 2022, Day 7 P3

March 21, 2023
number theorycongruence

Problem Statement

Let n=2k+1n = 2k + 1 be an odd positive integer, and mm be an integer realtively prime to nn{}. For each j=1,2,…,kj =1,2,\ldots,k we define pjp_j as the unique integer from the interval [−k,k][-k, k] congruent to m⋅jm\cdot j modulo nn{}. Prove that there are equally many pairs (i,j)(i,j) for which 1⩽i<j⩽k1\leqslant i<j\leqslant k which satisfy ∣pi∣>∣pj∣|p_i|>|p_j| as those which satisfy pipj<0p_ip_j<0.
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