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Part of 2017 China Second Round Olympiad

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2017 China Second Round Olympiad Test 2 Problem 4

Source: 2017 China Second Round Olympiad Test 2 Problem 4

9/10/2017
Let m,nm,n be integers greater than 1,m≥nm \geq n,a1,a2,…,ana_1,a_2,\dots,a_n are nn distinct numbers not exceed mm,which are relatively primitive.Show that for any real xx,there exists ii for which ∣∣aix∣∣≥2m(m+1)∣∣x∣∣||a_ix|| \geq \frac{2}{m(m+1)} ||x||,where ∣∣x∣∣||x|| denotes the distance between xx and the nearest integer to xx .
number theory