1998 Putnam Problems

    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 cone 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 on 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
     the product f (a) f '(a) f ''(a) f '''(a) ³ 0.

    A4. Let A₁ = 0, A₂ = 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 A₃ = A₂A₁ = 10, A₄ = A₃A₂ = 101. A₅ = A₄A₃ = 10110, and so forth. Determine all n such that 11 divides A_n.

    A5. Let F be a finite collection of open discs in R² whose union contains a set E Í R². Show that there is a pairwise disjoint subcollection D₁, D₂, ..., D_n in F such that E Í Èⁿ_i=1 3D_i. Here, if D is the disc of radius r and center P, then 3D is the disc or ratius 3r and center P.

    A6. Let A, B, C denote distinct points with integer coordinates in R². Prove that if (|AB| + |BC|)² < 8 |ABC| + 1, then A, B, C are 3 vertices of a square. Here |XY| is the length of segment XY and |ABC| is the area of triangle ABC.

    B1. Find the minimum value of
f = [(x + y)⁶ - (x⁶ + y⁶) - 2] / [(x + y)³ + (x³ + y³)] where y = 1/x, 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): x²+y²+z² = 1, z ³0}, C the unit circle {(x,y,0): x²+y²=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 a + B cos b, where A, B, a, and b are real numbers.

    B4. Find necessary and sufficient conditions on positive integers m and n so that
å _{i = 0}^mn-1 (-1) ^{f(i/m)+f(i/n)} = 0, where
for all real number x, f(x) = largest integer £ x.

    B5. Let N be the positive integer with 1998 decimal digits, all of them 1; that is
N = 111...1
with 1998 digits. Find the thousandth digit after the decimal point of ÖN.

    B6. Prove that, for any integers a, b, c, there exists a positive integer n such that
Ö (n³ + a n² + b n + c) is not an integer.