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Part of 2007 Putnam

Problems(2)

Putnam 2007 A2

Source:

12/3/2007
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola xy\equal{}1 and both branches of the hyperbola xy\equal{}\minus{}1. (A set S S in the plane is called convex if for any two points in S S the line segment connecting them is contained in S. S.)
Putnamgeometryconicshyperbolaanalytic geometryinequalitiesrectangle
Putnam 2007 B2

Source:

12/3/2007
Suppose that f:[0,1]→R f: [0,1]\to\mathbb{R} has a continuous derivative and that \int_0^1f(x)\,dx\equal{}0. Prove that for every α∈(0,1), \alpha\in(0,1), ∣∫0αf(x) dx∣≤18max⁡0≤x≤1∣f′(x)∣ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|
Putnamcalculusderivativeintegrationsymmetryfunctionalgebra