\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Weighted AM-GM. Let \(x_1, \dots, x_n > 0\) and \(\lambda_1, \dots, \lambda_n > 0\) with \(\sum_{k=1}^n \lambda_k = 1\). Show that $$ \sum_{k=1}^n \lambda_k x_k \ge \prod_{k=1}^n x_k^{\lambda_k}. $$
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