a maximum, minimum problem
TRANSCRIPT
PROBLEMS AND SOLUTIONS 77
A Covering Problem
Problem 82-2*, by D. J. NEWMAN (Temple University).Given any collection of squares with total area 3, prove that they can cover the
unit square. If the sides of the covering squares are all to be parallel to the sides ofthe unit square, then 3 is best possible. For the analogous problem with hypercubesin En, 3 is replaced by 2 1.
A Definite Integral
Problem 82-3, by A. J. JERRI (Clarkson College).Prove that
Jl(b(z,x))Jl(a(y,x)) Jl(b(y,z))dy 0<b<a,
(y,x) (y,z) (z,x)
where Jl(a (y, x))/(y, x) is the x-"generalized" J0-Hankel type translation of Jl(ay)/y,
Jl(a (y, x))_ayJl(ay)Jo(ax)- axJ(ax)Jo(ay)2 2(y,s) y -x
A Volume Problem
Problem 82-4, by M. K. LEWIS (Memorial University of Newfoundland).An asymmetrically positioned hole of radius b is drilled at right angles to the
axis of a solid right circular cylinder of radius a (a > b). If the distance between theaxis of the drill and the axis of the cylinder is p, determine the volume of materialdrilled out.
A Maximum, Minimum Problem
Problem 82-5, by T. SEKIGUCHI (University of Arkansas).In Euclidean space E3, with origin at O and coordinates (x i, x2, x3), let OX be
a ray in the closed first orthant, and ai be the angles between the positive xi axes and0X, i=1,2,3.
Determine the maximum and minimum values of
SOLUTIONS
A Polynomial Problem
Problem 80-16", by P. B. BORWEIN (The Mathematical Institute, Oxford, England).Does there exist a sequence of real algebraic polynomials {Pn (x)}, where Pn (x) is
of degree n, so that
lim measure {xlP’(x)/P(x)>-2n} 1?
One can show that
measure {xlP’,(x)/P,(x)>=2n} <= 1,
and L. Loomis (Bull. Amer. Math. Soc, 50 (1946), pp. 1082-1086) has shown that
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