MathDB

2

Part of 2012 Putnam

Problems(2)

Putnam 2012 A2

Source:

12/3/2012
Let ∗* be a commutative and associative binary operation on a set S.S. Assume that for every xx and yy in S,S, there exists zz in SS such that x∗z=y.x*z=y. (This zz may depend on xx and y.y.) Show that if a,b,ca,b,c are in SS and a∗c=b∗c,a*c=b*c, then a=b.a=b.
Putnamabstract algebragroup theorycollege contestsPutnam algebra
Putnam 2012 B2

Source:

12/3/2012
Let PP be a given (non-degenerate) polyhedron. Prove that there is a constant c(P)>0c(P)>0 with the following property: If a collection of nn balls whose volumes sum to VV contains the entire surface of P,P, then n>c(P)/V2.n>c(P)/V^2.
Putnamgeometryinequalities3D geometrysphereprismUSAMTS