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Real numbers such that a_1+a_2+...+a_n is an integer

Source: APMO 2006, Problem 1

March 24, 2006
number theory unsolvednumber theory

Problem Statement

Let nn be a positive integer. Find the largest nonnegative real number f(n)f(n) (depending on nn) with the following property: whenever a1,a2,...,ana_1,a_2,...,a_n are real numbers such that a1+a2+⋯+ana_1+a_2+\cdots +a_n is an integer, there exists some ii such that ∣ai−12∣≥f(n)\left|a_i-\frac{1}{2}\right|\ge f(n).
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