MathDB

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\equal{}1} \alpha^j_i \equal{} \sum^s_{i\equal{}1} \beta^j_i \forall j \equal{} \{1,2 \ldots, k\} we write \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\}. Prove that if \{\alpha_1, \alpha_2, \ldots , \alpha_s\} \overset{k}{\equal{}} \{\beta_1, \beta_2, \ldots , \beta_s\} and

s \leq k,

then there exists a permutation

\pi

\{1, 2, \ldots , s\}

such that \beta_i \equal{} \alpha_{\pi(i)} \forall i \equal{} 1,2, \ldots, s.

Problem Statement