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Almost all n,i have k dividing nCi

Source: 2018 China TST Day 2 Q2

January 2, 2018
number theorybinomial coefficientsDivisibility

Problem Statement

Given a positive integer kk, call nn good if among (n0),(n1),(n2),...,(nn)\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n} at least 0.99n0.99n of them are divisible by kk. Show that exists some positive integer NN such that among 1,2,...,N1,2,...,N, there are at least 0.99N0.99N good numbers.
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