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Upper bound for reciprocals of binomial coefficients

Source: VII Caucasus Mathematical Olympiad

March 13, 2022

binomial coefficientsalgebra

Problem Statement

Paul can write polynomial

(x+1)^n

, expand and simplify it, and after that change every coefficient by its reciprocal. For example if

n=3

Paul gets

(x+1)^3=x^3+3x^2+3x+1

and then

x^3+\frac13x^2+\frac13x+1

. Prove that Paul can choose

n

for which the sum of Paul’s polynomial coefficients is less than

2.022