International Mathematical Talent Search

**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. BarbeauMake sure that the front page of your solutions contains your

Department of Mathematics

University of Toronto

Toronto, ON M5S 3G3

These problems are made available through the quarterly journal

This page last updated: February 3, 1997

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

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