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A subset $X$ of $\mathbb{R}^n$ is convex if for all $x, y \in X$ and all $\lambda \in (0, 1)$, we have $\lambda x + (1- \lambda)y \in X$. If $X$ is a convex set, which of the following statements is necessarily TRUE?

  1. For every $ x \in X$ there exist $y, z \in X -\{x\}$ and $\lambda \in (0, 1)$ so that $x= \lambda y+ (1-\lambda ) z $
  2. If $x, y \in X$ and $\lambda \geq 0$, then $\lambda x + (1-\lambda)y  \in X$
  3. If $x_1, \dots , x_n \in X (n \geq 1)$, then $(x_1+ \dots + x_n)/n \in X$
  4. If $x \in X$, then $\lambda x \in X$ for all scalars $\lambda$
  5. If $x, y \in X$, then $x-y \in X$
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