MathDB

6

Part of 2007 IMC

Problems(2)

How many nonzero coefficients can a polynomial

Source: IMC 2007, Day 1, Problem 6

8/6/2007
How many nonzero coefficients can a polynomial P(x) P(x) have if its coefficients are integers and ∣P(z)∣≤2 |P(z)| \le 2 for any complex number z z of unit length?
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Sequence of polynomials eventually has real roots

Source: IMC 2007 Day 2 Problem 6

8/7/2007
Let f≠0 f \ne 0 be a polynomial with real coefficients. Define the sequence f0,f1,f2,… f_{0}, f_{1}, f_{2}, \ldots of polynomials by f0=f f_{0}= f and fn+1=fn+fn′ f_{n+1}= f_{n}+f_{n}' for every n≥0 n \ge 0. Prove that there exists a number N N such that for every n≥N n \ge N, all roots of fn f_{n} are real.
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