MathDB

40

Part of 1979 IMO Longlists

Problems(1)

Inequality regarding a polynomial

Source: ILL 1979-40

6/2/2011
A polynomial P(x)P(x) has degree at most 2k2k, where k=0,1,2,⋯k = 0, 1,2,\cdots. Given that for an integer ii, the inequality −k≤i≤k-k \le i \le k implies ∣P(i)∣≤1|P(i)| \le 1, prove that for all real numbers xx, with −k≤x≤k-k \le x \le k, the following inequality holds: ∣P(x)∣<(2k+1)(2kk)|P(x)| < (2k + 1)\dbinom{2k}{k}
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