Roots of equation

Let n \geq 1 and a_1, a_2 , \dots, a_n \in [0, 1]. Prove that

    \[\left | x-a_1 \right | + \left | x - a_2 \right | + \cdots + \left | x - a_n \right | = \frac{n}{2}\]

has solution in [0, 1].

Solution

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