Question 115
Let $a_1,a_2,\ldots,a_{2020}$ and $b_1,b_2,\ldots,b_{2020}$ be real numbers, not necessarily distinct. Suppose there are only finitely many positive integers $n$ for which
$$|a_1x-b_1|+|a_2x-b_2|+\cdots+|a_{2020}x-b_{2020}|=n$$
has exactly two real solutions. Prove that there are only finitely many positive integers $n$ for which this equation has at least one real solution.
$$|a_1x-b_1|+|a_2x-b_2|+\cdots+|a_{2020}x-b_{2020}|=n$$
has exactly two real solutions. Prove that there are only finitely many positive integers $n$ for which this equation has at least one real solution.