59th Putnam 1998

A1.  A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube?
A2.  Let C be the circle center (0, 0), radius 1. Let X, Y be two points on C with positive x and y coordinates. Let X1, Y1 be the points on the x-axis with the same x-coordinates as X and Y respectively, and let X2, Y2 be the points on the y-axis with the same y-coordinates. Show that the area of the region XYY1X1 plus the area of the region XYY2X2 depends only on the length of the arc XY, and not on its position.
A3.  Let R be the reals. Let f : R → R have a continuous third derivative. Show that there is a point a with f(a) f '(a) f ''(a) f '''(a) ≥ 0.
A4.  Define the sequence of decimal integers an as follows: a1 = 0; a2 = 1; an+2 is obtained by writing the digits of an+1 immediately followed by those of an. When is an a multiple of 11?
A5.  A finite collection of disks covers a subset X of the plane. Show that we can find a pairwise disjoint subcollection S, such that X ⊆ ∪{3D : D ∈ S}, where 3D denotes the disk with the same center as D and 3 times the radius.
A6.  P, Q, R are three (distinct) lattice points in the plane. Prove that if (PQ + QR)2 < 8 area PQR + 1, then P, Q, R are vertices of a square.
B1.  Find the minimum of { (x + 1/x)6 - (x6 + 1/x6) - 2 }/{ (x + 1/x)3 + (x3 + 1/x3) }, for x > 0.
B2.  Let P be the point (a, b) with 0 < b < a. Find Q on the x-axis and R on y = x, so that PQ + QR + RP is as small as possible.
B3.  Let S be the sphere center the origin and radius 1. Let P be a regular pentagon in the plane z = 0 with vertices on S. Find the surface area of the part of the sphere which lies above (z > 0) P or its interior.
B4.  For what m, n > 0 is ∑0mn-1 (-1)[i/m] + [i/n] = 0?
B5.  Let n be the decimal integer 11...1 (with 1998 digits). What is the 1000th digit after the decimal point of √n?
B6.  Show that for any integers a, b, c we can find a positive integer n such that n3 + a n2 + b n + c is not a perfect square.

To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 106 (1999) .  
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© John Scholes
4 Nov 1999