If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that: $a)$ $\mid c \mid \leq 1$ $b)$ $\mid a+c \mid \leq 1$ $c)$ $a^2+b^2+c^2 \leq 5$
Problem
Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016
Tags: algebra, inequalities