PROBLEMS FOR DECEMBER
Please send your solution to
Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than January 15, 2003.
It is important that your complete mailing address
and your email address appear on the front page.
Notes. An isosceles tetrahedron is one for which
the three pairs of oppposite edges are equal. For integers
$a$,
$b$ and
$n$,
$a\equiv b$, modulo
$n$, iff
$ab$ is a
multiple of
$n$.

192.

Let
$\mathrm{ABC}$ be a triangle,
$D$ be the midpoint of
$\mathrm{AB}$ and
$E$ a point on the side
$\mathrm{AC}$ for which
$\mathrm{AE}=2\mathrm{EC}$. Prove that
$\mathrm{BE}$ bisects the segment
$\mathrm{CD}$.

193.

Determine the volume of an isosceles tetrahedron for which
the pairs of opposite edges have lengths
$a$,
$b$,
$c$. Check your
answer independently for a regular tetrahedron.

194.

Let
$\mathrm{ABC}$ be a triangle with incentre
$I$. Let
$M$
be the midpoint of
$\mathrm{BC}$,
$U$ be the intersection of
$\mathrm{AI}$ produced with
$\mathrm{BC}$,
$D$ be the foot of the perpendicular from
$I$ to
$\mathrm{BC}$ and
$P$ be the foot of the perpendicular from
$A$ to
$\mathrm{BC}$. Prove that
$\Vert \mathrm{PD}\Vert \Vert \mathrm{DM}\Vert =\Vert \mathrm{DU}\Vert \Vert \mathrm{PM}\Vert \hspace{1em}.$

195.

Let
$\mathrm{ABCD}$ be a convex quadrilateral and let the midpoints
of
$\mathrm{AC}$ and
$\mathrm{BD}$ be
$P$ and
$Q$ respectively, Prove that
$\Vert \mathrm{AB}\Vert {}^{2}+\Vert \mathrm{BC}\Vert {}^{2}+\Vert \mathrm{CD}\Vert {}^{2}+\Vert \mathrm{DA}\Vert {}^{2}=\Vert \mathrm{AC}\Vert {}^{2}+\Vert \mathrm{BD}\Vert {}^{2}+4\Vert \mathrm{PQ}\Vert {}^{2}\hspace{1em}.$

196.

Determine five values of
$p$ for which the polynomial
${x}^{2}+2002x1002p$ has integer roots.

197.

Determine all integers
$x$ and
$y$ that satisfy
the equation
${x}^{3}+9\mathrm{xy}+127={y}^{3}$.

198.

Let
$p$ be a prime number and let
$f(x)$ be a polynomial
of degree
$d$ with integer coefficients such that
$f(0)=0$ and
$f(1)=1$ and that, for every positive integer
$n$,
$f(n)\equiv 0$ or
$f(n)\equiv 1$, modulo
$p$. Prove that
$d\ge p1$. Give an example of such a polynomial.