M. S. Klamkin

Vector Proofs in Solid Geometry

(1970). Vector Proofs in Solid Geometry. The American Mathematical Monthly: Vol. 77, No. 10, pp. 1051-1065.

On best quadratic triangle inequalities

AbstractContrary to published results, it is shown that there do not exist ‘strongest’ (or ‘best possible’) homogeneous quadratic polynomial triangle inequalities of the form

A Note on an nth Order Linear Differential Equation

q(R,r) \leqslant s^2 \leqslant Q(R,r)

Transformation of boundary value problems into initial value problems

without further restrictions. Also, several best inequalities for symmetric functions of three positive variables are considered.

Inequalities for Sums of Distances

Abstract Three types of problems in gravitational attraction are considered. In the first type, two isoperimetric ones are solved elementarily by means of a known inequality concerning integral means. In the second one, it is shown that if a homogeneous ellipsoid is divided into two measurable sets, then the gravitational attractive force between them is greatest when the two sets are hemiellipsoids formed by a plane containing the two largest principal axes. In the third one, it is shown that the motion of a free particle in a straight tunnel through a homogeneous ellipsoidal planet is simple harmonic. Furthermore, the periods achieve their maximum and minimum values when the tunnels are parallel to the largest and smallest axes, respectively.

On the teaching of mathematics so as to be useful

(1) (x-a)n(x-b)nDny = Ay, (a4 b), has been solved previously by Halphen (see Kamke, Differentialgleichungen, p. 541) by means of the substitution y = (x-b)n-)xF(loga) which transforms Eq. (1) into a linear one in F with constant coefficients. We effect the solution here in a much simpler way. Also, we find the limit of the solution as b -+ a. We assume a solution of the form y = (x-a)m(x-6b)n-m,. On substituting back in Eq. (1) and applying Liebnizs rule for the differentiation of a product, we find that m must be any root (mr) of the nth order polynomial equation

Simultaneous Generalizations of the Theorems of Ceva and Menelaus

In the well known Buffon needle problem, a needle of length L is dropped on a board ruled with equidistant parallel lines of spacing D where D?L; it is required to determine the probability that the needle will intersect one of the lines. B3arbiers [1] elegant method was to let this problem depend on the following one: What is the mathematical expectation of the number of points of intersection when a polygonal line (convex or not) is thrown upon the board? The perimeter of the polygonal line can be subdivided into IV rectilinear parts a,, a2, ... , aN, all less than D. Associated with these IV parts are the variables x1, x2, , x, x,N such that