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5

Part of 1999 Putnam

Problems(2)

Putnam 1999 A5

Source:

12/22/2012
Prove that there is a constant CC such that, if p(x)p(x) is a polynomial of degree 19991999, then ∣p(0)∣≤C∫−11∣p(x)∣ dx.|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.
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Putnam 1999 B5

Source:

12/22/2012
For an integer n≥3n\geq 3, let θ=2π/n\theta=2\pi/n. Evaluate the determinant of the n×nn\times n matrix I+AI+A, where II is the n×nn\times n identity matrix and A=(ajk)A=(a_{jk}) has entries ajk=cos⁡(jθ+kθ)a_{jk}=\cos(j\theta+k\theta) for all j,kj,k.
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