Question 64
A capitalist returns from a business trip and brings $n$ gifts for his $n$ children. For $i \in \{1, 2, \dots, n\}$, his $i$-th oldest child considers $x_i$ of these items to be desirable. Assume that the numbers $x_1, \dots, x_n$ are positive and satisfy
$$
\frac{1}{x_1} + \dots + \frac{1}{x_n} \le 1.
$$
Prove that the children may distribute the gifts among themselves in such a way that each child receives a gift that it likes.
$$
\frac{1}{x_1} + \dots + \frac{1}{x_n} \le 1.
$$
Prove that the children may distribute the gifts among themselves in such a way that each child receives a gift that it likes.