NMO Selection Tests for the Balkan and International Mathematical Olympiads

Let $$f(x_1,x_2,\dots ,x_n)=\max \left\{\frac{x_k}{1+x_1+x_2+\cdots +x_k}:k=1,2,\dots ,n\right\},$$ where $x_1\ge 0,x_2\ge 0,\dots ,x_n\ge 0$ and $x_1+x_2+\cdots +x_n=1$, and notice that $$\frac{a_1}{1+a_1}=\frac{a_2}{1+a_1+a_2}=\cdots =\frac{a_n}{1+a_1+a_2+\cdots +a_n}$$ for a unique $n$-tuple $(a_1,a_2,\dots ,a_n)$ of non-negative real numbers which add up to 1, namely, $a_k=2^{k/n}-2^{(k-1)/n}$, $k=1,2,\dots ,n$, in which case $f(a_1,a_2,\dots ,a_n)=1-2^{-1/n}$. Now let $(x_1,x_2,\dots ,x_n)\ne (a_1,a_2,\dots ,a_n)$, where the $x_i$ are non- negative real numbers which add up to 1. Since $x_1+x_2+\cdots +x_n=a_1+a_2+\cdots +a_n$, it follows that $x_k>a_k$ for some index $k$. Let $m=\min \{k:x_k>a_k\}$. Then $x_k\le a_k$, $k<m$, so $$\begin{gathered} f(x_1,x_2,\dots ,x_n)\ge \frac{x_m}{1+x_1+x_2+\cdots +x_m} \\ >\frac{a_m}{1+a_1+a_2+\cdots +a_m}=f(a_1,a_2,\dots ,a_n). \end{gathered}$$ Consequently, the required minimum is $1-2^{-1/n}$.