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China Mathematical Olympiad 2020 Q3

Source: Changsha

November 24, 2020
number theoryinequalitiesalgebraChinaBashing

Problem Statement

Let nn be positive integer such that there are exactly 36 different prime numbers that divides n.n. For k=1,2,3,4,5,k=1,2,3,4,5, cnc_n be the number of integers that are mutually prime numbers to nn in the interval [(k−1)n5,kn5].[\frac{(k-1)n}{5},\frac{kn}{5}] . c1,c2,c3,c4,c5c_1,c_2,c_3,c_4,c_5 is not exactly the same.Prove that∑1≤i<j≤5(ci−cj)2≥236.\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.
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