ISL 1989 Problem 69

Let $k$ and $s$ be positive integers. For sets of real numbers $\{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $\{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy

$$\sum^s_{i=1} \alpha^j_i = \sum^s_{i=1} \beta^j_i \forall j = \{1,2 \ldots, k\}$$

we write $$\{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{=} \{\beta_1, \beta_2, \ldots , \beta_s\}.$$

Prove that if $$\{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{=} \{\beta_1, \beta_2, \ldots , \beta_s\}$$ and $s \leq k,$ then there exists a permutation $\pi$ of $\{1, 2, \ldots , s\}$ such that

$$\beta_i = \alpha_{\pi(i)} \forall i = 1,2, \ldots, s.$$