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Fifty-ninth Annual
William Lowell Putnam
Mathematical Competition
Saturday, December 5, 1998

Examination A

Problem A1

A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side length of the cube?

Problem A2

Let s be any arc of the unit circle lying entirely in the first quadrant. Let A be the area of the region lying below s and above the x-axis and let B be the area lying to the right of the y-axis adn to the left of s. Prove that A+B depends only on the arc length, and not on the position, of s.

Problem A3

let f be a real function on the real line with continuous third derivative. Prove that there exists a point a such that

Problem A4

Let A₁ = 0 and A₂= 1. For n > 2, the number A_n is defined by concatenating the decimal expansions of A_n-1 and A_n-2 fro left to right. For example, A₃ = A₂A₁ = 10, A₄ = A₃A₂ = 101, A₅ = A₄A₃ = 10110, and so forth. Determine all n such that 11 divides A_n.

Problem A5

Left be a finite collection of open disks in R² whose union contains a set . Show that there is a pairwise disjoint subcollection in such that

Here, if D is the disk of radius r and center P, then 3D is the disk of radius 3r and center P.

Problem A6

Let A, B, and C denote distinct points with integer coordinates in R². Prove that if

(|AB| + |BC|)² < 8[ABC] + 1

then A, B and C are three vertices of a square. Here |XY| is the length of the segment XY and [ABC] is the area of the triangle ABC.

Examination B

Problem B1

Find the minimum value of

Problem B2

Given a point (a,b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a,b), one on the x-axis, and one on the line y=x. You may assume that a triangle of minimum perimeter exists.

Problem B3

Let H be the unit hemisphere ,C the unit circle , and P a regular pentagon inscribed in C. Determine the surface area of that portion of H lying ove the planar region inside P, and write your answer in the form , where A, B, and are real numbers.

Problem B4

Find necessary and sufficient conditions on positive integers m and n so that

Problem B5

Let N be the positive integer with 1998 digits, all of them 1. Find the thousandth digint after the decimal point of .

Problem B6

Prove that, for any integers a, b and c, there exists a positive integer n such that is not an integer.

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Allen D Bell
7/10/2000