Find all real polynomials p(x) of degree n≥2 for which there exist real numbers r1<r2<...<rn such that
(i) p(r_i) \equal{} 0, 1 \le i \le n, and
(ii) p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1.
Follow-up: In terms of n, what is the maximum value of k for which k consecutive real roots of a polynomial p(x) of degree n can have this property? (By "consecutive" I mean we order the real roots of p(x) and ignore the complex roots.) In particular, is k \equal{} n \minus{} 1 possible for n≥3? Putnamalgebrapolynomialcollege contests