Problem 1

Let

n

be a positive integer. Find the largest nonnegative real number

f(n)

such that whenever real numbers

a_1,a_2,\ldots,a_n

have an integer sum, there is an index

i

for which

\left|a_i-\frac12\right|\ge f(n)

Step 3 of 5: Use the half-integer obstruction

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\sum_{i=1}^n a_i\in\mathbb Z,\quad\frac n2\in\mathbb Z+\frac12

Detailed analysis

The sum $\sum_{i=1}^n a_i$ is an integer, whereas $n/2$ is a half-integer because $n$ is odd. Their distance is therefore at least $\frac12$ , contradicting the strict bound in the preceding step. Hence some index satisfies $\left|a_i-\frac12\right|\ge\frac1{2n}$ .