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Part of 1998 Putnam

Problems(2)

1998 Putnam A3

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4/19/2013
Let ff be a real function on the real line with continuous third derivative. Prove that there exists a point aa such that f(a)⋅f′(a)⋅f′′(a)⋅f′′′(a)≥0.f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.
Putnamfunctionintegrationcollege contestsPutnam calculus
Putnam 1998 B3

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10/28/2012
Let HH be the unit hemisphere {(x,y,z):x2+y2+z2=1,z≥0}\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}, CC the unit circle {(x,y,0):x2+y2=1}\{(x,y,0):x^2+y^2=1\}, and PP the regular pentagon inscribed in CC. Determine the surface area of that portion of HH lying over the planar region inside PP, and write your answer in the form Asin⁡α+Bcos⁡βA \sin\alpha + B \cos\beta, where A,B,α,βA,B,\alpha,\beta are real numbers.
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