MathLabs

Problem 1

Let nn be a positive integer. Find the largest nonnegative real number f(n)f(n) such that whenever real numbers a1,a2,…,ana_1,a_2,\ldots,a_n have an integer sum, there is an index ii for which ∣ai−12∣≥f(n)\left|a_i-\frac12\right|\ge f(n).
Step 1 of 5: Dispose of the even case
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n even:a1=⋯=an=12n\text{ even}:\quad a_1=\cdots=a_n=\frac12
Detailed analysis

If nn is even and every ai=12a_i=\frac12, then the sum is the integer n/2n/2, while every distance ∣ai−12∣\left|a_i-\frac12\right| is zero. Thus f(n)=0f(n)=0 for even nn.