PROBLEMS FOR DECEMBER
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
$\sum _{i\ne j}{a}_{i}{b}_{j}=0\hspace{1em}.$
Prove that
$\sum _{i\ne j}{b}_{i}{b}_{j}<0\hspace{1em}.$

122.

Determine all functions
$f$ from the real numbers
to the real numbers that satisfy
$f(f(x)+y)=f({x}^{2}y)+4f(x)y$
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
$\frac{({1}^{4}+\frac{1}{4})({3}^{4}+\frac{1}{4})({5}^{4}+\frac{1}{4})\dots ({11}^{4}+\frac{1}{4})}{({2}^{4}+\frac{1}{4})({4}^{4}+\frac{1}{4})({6}^{4}+\frac{1}{4})\dots ({12}^{4}+\frac{1}{4})}=\frac{1}{313}\hspace{1em}.$

125.

Determine the set of complex numbers
$z$ which
satisfy
$\mathrm{Im}\hspace{1em}({z}^{4})=(\mathrm{Re}\hspace{1em}({z}^{2})){}^{2}\hspace{1em},$
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
$n$.