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International Mathematical Talent Search

# Round 18

Problem 1/18. Determine the minimum length of the interval [a,b] such that for all real numbers for which 19x + 95y = 1995.

Problem 2/18. For a positive integer , let P(n) denote the product of the positive integer divisors (including 1 and n) of n. Find the smallest n for which .

Problem 3/18. The graph shown below has 10 vertices, 15 edges, and each vertex is of order 3 (i.e., at each vertex 3 edges meet). Some of the edges are labelled 1, 2, 3, 4, 5 as shown. Prove that it is possible to label the remaining edges 6, 7, 8, . . . , 15 so that at each vertex the sum of the labels on the edges meeting at that vertex is the same.

Problem 4/18. Let a, b, c, d be distinct real numbers such that a+b+c+d = 3 and . Find the value of the expression

Problem 5/18. Let a and b be two lines in the plane, and let C be a point, as shown below. Using only a compass and an unmarked straightedge, construct an isosceles right triangle ABC, so that A is on line a, B is on line b, and AB is the hypotenuse of triangle ABC.

Solve as many of the problems as you can (you need not solve them all), and mail your solutions to:

Professor E. J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
These problems are made available through the quarterly journal Mathematics and Informatics. Student subscriptions at US\$12 (student rate) or US\$18 (teacher rate) may be ordered from Prof. George Berzsenyi, Department of Mathematics, Box 121, Rose-Hulman Institute of Technology, Terre Haute, IN 47803-3999, U.S.A. Remittances should be payable to Mathematics and Informatics.