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 5 of 5: State the answer
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f(n)={0,n even,12n,n odd,f(n)=\begin{cases}0,&n\text{ even},\\\frac1{2n},&n\text{ odd},\end{cases}
Detailed analysis

Combining the even construction with the odd-case necessity and sharp example gives f(n)=0f(n)=0 for even nn and f(n)=12nf(n)=\frac1{2n} for odd nn.