FIFTY-NINTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1998 Examination A

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?

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 of the region lying to the right of the y-axis and to the left of s. Prove that A + B depends only on the arc length, and not of the position, of s.

A3. Let f be a real function on the real line with continuous third derivative. Prove that there exists a point a such that

f(a) · f′(a) · f′′(a) · f′′′(a) ≥ 0.

A4. Let A_{1} = 0 and A_{2} = 1. For n ≥ 2, the number A_{n} is defined by concatenating the decimal expansions of A_{n−1} and A_{n−2} from left to right. For example A_{3} = A_{2}A_{1} = 10, a_{4} = A_{3}A_{2} = 101, A_{5} = A_{4}A_{3} = 10110, and so forth. Determine all n such that 11 divides A_{n}.

A5. Let F be a finite collection of open discs in R^{2} whose union contains a set E ⊆ R^{2}. Show that there is a pairwise disjoint subcollection D_{1}, …, D_{n} in F such that

∪_{j=1}^{n}3D_{j} ⊇ E.

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

A6. Let A, B, C denote distinct points with integer coordinates in R^{2}. Prove that if

(|AB| + |BC|)^{2} < 8 · [ABC] + 1

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

FIFTY-NINTH ANNUAL WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

Saturday, December 5, 1998 Examination B

B1. Find the minimum value of

\frac{(x + 1/x)^{6} − (x^{6} + 1/x^{6}) − 2}{(x + 1/x)^{3} + (x^{3} + 1/x^{3})}

for x > 0.

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.

B3. Let H be the unit hemisphere {(x, y, z) : x^{2} + y^{2} + z^{2} = 1, z ≥ 0}, C the unite circle {(x, y, 0 : x^{2} + y^{2} = 1}, and P a regular pentagon inscribed in C. Determine the surface area of that portion of H lying over the planar region inside P, and write your answer in the form A sin α + B cos β, where A, B, α, β are real numbers.

B4. Find necessary and sufficient conditions on positive integers m and n so that

∑_{i=0}^{mn−1}(−1)^{⌊i/m⌋+⌊i/n⌋} = 0.

B5. Let N be the positive integer with 1998 digits, all of them 1; tht is,

N = 1111 ⋯ 11.

Find the thousandth digit after the decimal point of \sqrt{N}.

B6. Prove that, for any integers a, b, c, there exists a positive integer n such that \sqrt{n^{3} + an^{2} + bn + c} is not an integer.