I have a great problem from Qvant magazine I can’t solve. Please help!
Suppose that for any $-1\leq x\leq 1$, $|ax^2+bx+c|\leq 1$. Find the maximum possible value of $|a|+|b|+|c|$.
It is suffices to find the maximum value of $|a|+|b|+|c|$ if for any $-1\leq x\leq 1$, $(ax^2+bx+c)^2\leq 1$. I inserted $x=-1, x=0, x=1$, and I got that $c\leq 1, a+c\leq 1, a+b+c\leq 1$. How to continue? Or maybe it is not ppssoble to solve the problem with my attempts? Any solution?