Take $p_1, p_2, \ldots, p_n$ points in $\mathbb{Q}^m \subset \mathbb{R}^m$, $n>m$, such that $0$ is in the interior of the convex hull $conv(p_1, \ldots, p_n)$.

Prove that exists $\lambda_1, \lambda_2, \ldots, \lambda_n \in \mathbb{Q}$, $\lambda_i \geq0$, such that $\sum_{i=1}^{n}\lambda_i = 1$ and $$\sum_{i=1}^{n}\lambda_i p_i = 0.$$

The answer is pretty clear for real coefficients, but I am having trouble trying to prove that exists rational coefficients.


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