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Part of 1999 Putnam

Problems(2)

Putnam 1999 A2

Source:

12/22/2012
Let p(x)p(x) be a polynomial that is nonnegative for all real xx. Prove that for some kk, there are polynomials f1(x),f2(x),…,fk(x)f_1(x),f_2(x),\ldots,f_k(x) such that p(x)=∑j=1k(fj(x))2.p(x)=\sum_{j=1}^k(f_j(x))^2.
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Putnam 1999 B2

Source:

12/22/2012
Let P(x)P(x) be a polynomial of degree nn such that P(x)=Q(x)P′′(x)P(x)=Q(x)P^{\prime\prime}(x), where Q(x)Q(x) is a quadratic polynomial and P′′(x)P^{\prime\prime}(x) is the second derivative of P(x)P(x). Show that if P(x)P(x) has at least two distinct roots then it must have nn distinct roots.
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