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Problems

Please send solutions to

E.J. Barbeau

Department of Mathematics

University of Toronto

Toronto, ON M5S 3G3

no later than December 31, 2001. Please make sure that your name, email address and complete mailing address are on the first page.

115.: Let $U$ be a set of $n$ distinct real numbers and let $V$ be the set of all sums of distinct pairs of them, i.e.,

V = {x + y : x, y \in U, x \neq y} .

What is the smallest possible number of distinct elements that

V

can contain?

x^{4} + 5 x^{3} + 6 x^{2} - 4 x - 16 = 0

has exactly two real solutions.

(a - 1) (\frac{1}{\sin x} + \frac{1}{\cos x} + \frac{1}{\sin x \cos x}) = 2 .

a^{2} (b + c - a) + b^{2} (c + a - b) + c^{2} (a + b - c) \leq 3 abc .

When does equality hold?

‖ AB ‖^{2} + ‖ BC ‖^{2} + ‖ CA ‖^{2} = 3 (‖ GA ‖^{2} + ‖ GB ‖^{2} + ‖ GC ‖^{2}) .

120.: Determine all pairs of nonnull vectors x, y for which the following sequence ${a_{n} : n = 1, 2, \dots}$ is (a) increasing, (b) decreasing, where

a_{n} = ‖ x - n y ‖ .