Problem 1372

1. Given the sets of real numbers $\{\alpha_1, \alpha_2, \ldots , \alpha_s\}$ and $\{\beta_1, \beta_2, \ldots, \beta_s\}$ that satisfy:
$$\sum_{i=1}^s \alpha_i^j = \sum_{i=1}^s \beta_i^j \quad \forall j = \{1, 2, \ldots, k\}$$
we need to prove that if $s \leq k$, then there exists a permutation $\pi$ of $\{1, 2, \ldots, s\}$ such that:
$$\beta_i = \alpha_{\pi(i)} \quad \forall i = 1, 2, \ldots, s.$$

2. Consider the polynomial $P(x)$ defined by the roots $\alpha_1, \alpha_2, \ldots, \alpha_s$:
$$P(x) = \prod_{i=1}^s (x - \alpha_i)$$
Similarly, define the polynomial $Q(x)$ with roots $\beta_1, \beta_2, \ldots, \beta_s$:
$$Q(x) = \prod_{i=1}^s (x - \beta_i)$$

3. The sums of powers of the roots of a polynomial are related to the coefficients of the polynomial. Specifically, the elementary symmetric polynomials in the roots can be expressed in terms of the sums of powers of the roots using Newton's identities.

4. Given that:
$$\sum_{i=1}^s \alpha_i^j = \sum_{i=1}^s \beta_i^j \quad \forall j = \{1, 2, \ldots, k\}$$
and $s \leq k$, we can use these sums to determine the coefficients of the polynomials $P(x)$ and $Q(x)$.

5. Since the sums of powers of the roots are equal for all $j = 1, 2, \ldots, k$, the coefficients of $P(x)$ and $Q(x)$ must be identical. Therefore, the polynomials $P(x)$ and $Q(x)$ are identical:
$$\prod_{i=1}^s (x - \alpha_i) = \prod_{i=1}^s (x - \beta_i)$$

6. If two polynomials with real coefficients are identical, their roots must be identical up to a permutation. Hence, there exists a permutation $\pi$ of $\{1, 2, \ldots, s\}$ such that:
$$\beta_i = \alpha_{\pi(i)} \quad \forall i = 1, 2, \ldots, s.$$

$\blacksquare$