IMO 1989 LL PHI76

Origin: PHI

Problem

Let k and s be positive integers. For sets of real numbers {\alpha1, \alpha2, . . . , \alphas} and {\beta1, \beta2, . . . , \betas} that satisfy s i=1 \alphaj i = s i=1 \betaj i for each j = 1, 2, . . ., k, we write {\alpha1, \alpha2, . . . , \alphas} =k {\beta1, \beta2, . . . , \betas}. Prove that if {\alpha1, \alpha2, . . . , \alphas} =k {\beta1, \beta2, . . . , \betas} and s \leqk, then there ex- ists a permutation \pi of {1, 2, . . ., s} such that \betai = \alpha\pi(i) for i = 1, 2, . . . , s.