a maximum, minimum problem

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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