5

Part of 1998 IMC

Problems(2)

polynomial with real coefficients

Source: IMC 1998 day 1 problem 5

P

be a polynomial of degree

n

with only real zeros and real coefficients. Prove that for every real

x

(n-1)(P'(x))^2\ge nP(x)P''(x)

. When does equality occur?

algebrapolynomialinductionreal analysisreal analysis unsolved

IMC 1998 Problem 11

Source: IMC 1998 Day 2 Problem 5

S

is a family of balls in

\mathbb{R}^{n}

n > 1

) such that the intersection of any two contains at most one point. Show that the set of points belonging to at least two members of

S