Summary: Let \(A\) be a nonnegative symmetric \(5 \times 5\) matrix with eigenvalues \(\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5\). We show that if \(\sum_{i = 1}^5 \lambda_i \geq \frac{1}{2} \lambda_1\) then \(\lambda_3 \leq \sum_{i = 1}^5 \lambda_i\). J. J. McDonald and M. Neumann [Contemp. Math. 259, 387–407 (2000; Zbl 0965.15009)] showed that \(\lambda_1 + \lambda_3 + \lambda_4 \geq 0\). Let \(\sigma = (\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5)\) be a list of decreasing real numbers satisfying: (1) \(\sum_{i = 1}^5 \lambda_i \geq \frac{1}{2} \lambda_1\), (2) \(\lambda_3 \leq \sum_{i = 1}^5 \lambda_i\), (3) \(\lambda_1 + \lambda_3 + \lambda_4 \geq 0\), (4) the Perron property, that is \(\lambda_1 = \max_{\lambda \in \sigma} | \lambda |\). We show that \(\sigma\) is the spectrum of a nonnegative symmetric \(5 \times 5\) matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problem for \(n = 5\) in a region for which a solution has not been known before.