BG-MO 1975 Problem 3

Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ( $a_0\ne0$ ) such that $|f(x)|\le1$ for every $x\in[-1,1]$ . Prove that

(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which (b) $|a_0|\le4$ .

V. Petkov