62nd Ukrainian National Mathematical Olympiad

Let the numbers be arranged in non-increasing order: $x_1\ge x_2\ge \cdots \ge x_n$. Consider the sums $2x_1\ge x_1+x_2\ge x_1+x_3\ge \cdots \ge x_1+x_n>x_1$. Therefore, some two of the sums $x_1+x_2,x_1+x_3,\dots ,x_1+x_n$ differ by no more than a factor of $\sqrt[4]{2}$.