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# Round 18

Problem 1/18. Determine the minimum length of the interval [a,b] such that a <= x+y <= b for all real numbers x >= y >= 0 for which 19x + 95y = 1995.

Problem 2/18. For a positive integer n >= 2, let P(n) denote the product of the positive integer divisors (including 1 and n) of n. Find the smallest n for which P(n) = n^(10).

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.

*
/|\
/ 3 \
/  |  \
/   *   \        *    = vertex
/   / \   \       ---  = edge
*   /   \   *
|\ /     \ /|
4 /       \ 2
|/ \     / \|
*   *   *   *
\  |\ /|  /
\ 5 \ 1 /
\|/ \|/
*   *

Problem 4/18. Let a, b, c, d be distinct real numbers such that a+b+c+d = 3 and a^2 + b^2 + c^2 + d^2 = 45. Find the value of the expression

5                 5                 5                 5
a                 b                 c                 d
--------------- + --------------- + --------------- + --------------- .
(a-b)(a-c)(a-d)   (b-a)(b-c)(b-d)   (c-a)(c-b)(c-d)   (d-a)(d-b)(d-c)

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.

*
*
*   a
*
*
*

* C

***************
b

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