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Part of 1996 Putnam

Problems(2)

Putnam 1996 A2

Source:

6/4/2014
Let C1\mathcal{C}_1 and C2\mathcal{C}_2 be circles whose centers are 1010 units apart, and whose radii are 11 and 33. Find, with proof, the locus of all points MM for which there exists points X∈C1,Y∈C2X\in \mathcal{C}_1,Y\in \mathcal{C}_2 such that MM is the midpoint of XYXY.
Putnamcollege contests
Putnam 1996 B2

Source:

6/6/2014
Prove the inequality for all positive integer nn : (2n−1e)2n−12<1⋅3⋅5⋯(2n−1)<(2n+1e)2n+12 \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}}
Putnaminequalitiesinductionlogarithmsgeometryrectangleintegration