PROBLEMS FOR APRIL
The first five problems appeared on the annual University of
Toronto Undergraduate Competition.
Please send your solutions to
Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than May 15, 2002.

139.

Let
$A$,
$B$,
$C$ be three pairwise orthogonal
faces of a tetrahedran meeting at one of its vertices and
having respective areas
$a$,
$b$,
$c$. Let the face
$D$
opposite this vertex have area
$d$. Prove that
${d}^{2}={a}^{2}+{b}^{2}+{c}^{2}\hspace{1em}.$

140.

Angus likes to go to the movies. On Monday,
standing in line, he noted that the fraction
$x$ of the
line was in front of him, while
$1/n$ of the line was behind
him. On Tuesday, the same fraction
$x$ of the line was
in front of him, while
$1/(n+1)$ of the line was behind him.
On Wednesday, the same fraction
$x$ of the line was in front
of him, while
$1/(n+2)$ of the line was behind him.
Determine a value of
$n$ for which this is possible.

141.

In how many ways can the rational
$2002/2001$
be written as the product of two rationals of the form
$(n+1)/n$, where
$n$ is a positive integer?

142.

Let
$x,y>0$ be such that
${x}^{3}+{y}^{3}\le xy$.
Prove that
${x}^{2}+{y}^{2}\le 1$.

143.

A sequence whose entries are
$0$ and
$1$ has the
property that, if each
$0$ is replaced by
$01$ and each
$1$
by
$001$, then the sequence remains unchanged. Thus, it starts
out as
$010010101001\dots $. What is the
$2002$th term of the
sequence?

144.

Let
$a$,
$b$,
$c$,
$d$ be rational numbers for which
$\mathrm{bc}\ne \mathrm{ad}$. Prove that there are infinitely many rational values
of
$x$ for which
$\sqrt{(a+\mathrm{bx})(c+\mathrm{dx})}$ is rational. Explain the
situation when
$\mathrm{bc}=\mathrm{ad}$.