PROBLEMS FOR DECEMBER

Canadian Mathematical Society

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PROBLEMS FOR DECEMBER

Please send your solutions to

Professor E.J. Barbeau

Department of Mathematics

University of Toronto

Toronto, ON M5S 3G3

no later than January 31, 2002.

Note. The incentre of a triangle is the centre of the inscribed circle that touches all three sides. A set is connected if, given two points in the set, it is possible to trace a continuous path from one to the other without leaving the set.

121.: Let $n$ be an integer exceeding 1. Let $a_{1}, a_{2}, \dots, a_{n}$ be posive real numbers and $b_{1}, b_{2}, \dots, b_{n}$ be arbitrary real numbers for which

\sum_{i \neq j} a_{i} b_{j} = 0 .

Prove that

\sum_{i \neq j} b_{i} b_{j} < 0 .

122.: Determine all functions $f$ from the real numbers to the real numbers that satisfy

f (f (x) + y) = f (x^{2} - y) + 4 f (x) y

for any real numbers

x

y

123.: Let $a$ and $b$ be the lengths of two opposite edges of a tetrahedron which are mutually perpendicular and distant $d$ apart. Determine the volume of the tetrahedron.

124.: Prove that

\frac{(1^{4} + \frac{1}{4}) (3^{4} + \frac{1}{4}) (5^{4} + \frac{1}{4}) \dots (11^{4} + \frac{1}{4})}{(2^{4} + \frac{1}{4}) (4^{4} + \frac{1}{4}) (6^{4} + \frac{1}{4}) \dots (12^{4} + \frac{1}{4})} = \frac{1}{313} .

125.: Determine the set of complex numbers $z$ which satisfy

Im (z^{4}) = (Re (z^{2}))^{2},

and sketch this set in the complex plane. (Note: Im and Re refer respectively to the imaginary and real parts.)

126.: Let $n$ be a positive integer exceeding $1$ , and let $n$ circles (i.e., circumferences) of radius 1 be given in the plane such that no two of them are tangent and the subset of the plane formed by the union of them is connected. Prove that the number of points that belong to at least two of these circles is at least $n$ .

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