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Abstract
MATHEMATICS MAGAZINE 1993. Proposed by Kimberly D. Apple, Columbus State University, GA. Each face of an icosahedron is colored blue or white in such a way that any blue face is adjacent to no more than two other blue faces. What is the maximum number of blue faces? (Two faces are considered adjacent if they share an edge.) 1994. Proposed by Donald E. Knuth, Computer Science Department, Stanford University, CA. Let X0 = 0, and suppose Xn+1 is equally likely to be either Xn + 1 or Xn − 2. What is the probability, pm , that Xn ≤ m for all n ≥ 0? 1995. Proposed by Michel Bataille, Rouen, France. Let a1, a2, . . . , an be distinct positive real numbers (n > 2). For j = 1, 2, . . . , n, let s j = ∑ i = j ai . Consider the n × n matrix M = (mi, j ) defined by mi,i = 0 and mi, j = ai/s j for 1 ≤ i < j ≤ n. Show that there exist column vectors X and Y such that X T Y = 1 and Y X T = lim k→∞ M .