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B5

Part of 2011 Putnam

Problems(1)

Putnam 2011 B5

Source:

12/5/2011
Let a1,a2,…a_1,a_2,\dots be real numbers. Suppose there is a constant AA such that for all n,n, ∫−∞∞(∑i=1n11+(x−ai)2)2 dx≤An.\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An. Prove there is a constant B>0B>0 such that for all n,n, ∑i,j=1n(1+(ai−aj)2)≥Bn3.\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.
PutnamintegrationfunctioninequalitiescalculusSupportvector