PROBLEMS FOR MAY
Please send your solutions to
E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3.
no later then June 30, 2002.

145.

Let
$\mathrm{ABC}$ be a right triangle with
$\angle A={90}^{\u02c6}$.
Let
$P$ be a point on the hypotenuse
$\mathrm{BC}$, and let
$Q$ and
$R$
be the respective feet of the perpendiculars from
$P$ to
$\mathrm{AC}$ and
$\mathrm{AB}$. For what position of
$P$ is the length of
$\mathrm{QR}$ minimum?

146.

Suppose that
$\mathrm{ABC}$ is an equilateral triangle.
Let
$P$ and
$Q$ be the respective midpoint of
$\mathrm{AB}$ and
$\mathrm{AC}$,
and let
$U$ and
$V$ be points on the side
$\mathrm{BC}$ with
$4\mathrm{BU}=4\mathrm{VC}=\mathrm{BC}$ and
$2\mathrm{UV}=\mathrm{BC}$. Suppose that
$\mathrm{PV}$ are joined and
that
$W$ is the foot of the perpendicular from
$U$ to
$\mathrm{PV}$ and
that
$Z$ is the foot of the perpendicular from
$Q$ to
$\mathrm{PV}$.


Explain how that four polygons
$\mathrm{APZQ}$,
$\mathrm{BUWP}$,
$\mathrm{CQZV}$ and
$\mathrm{UVW}$ can be rearranged to form a rectangle. Is this rectangle
a square?

147.

Let
$a>0$ and let
$n$ be a positive integer.
Determine the maximum value of
$\frac{{x}_{1}{x}_{2}\dots {x}_{n}}{(1+{x}_{1})({x}_{1}+{x}_{2})\dots ({x}_{n1}+{x}_{n})({x}_{n}+{a}^{n+1})}$
subject to the constraint that
${x}_{1},{x}_{2},\dots ,{x}_{n}>0$.

148.

For a given prime number
$p$, find the number of
distinct sequences of natural numbers (positive integers)
$\{{a}_{0},{a}_{1},\dots ,{a}_{n}\dots \}$ satisfying, for each
positive integer
$n$, the equation
$\frac{{a}_{0}}{{a}_{1}}+\frac{{a}_{0}}{{a}_{2}}+\dots +\frac{{a}_{0}}{{a}_{n}}+\frac{p}{{a}_{n+1}}=1\hspace{1em}.$

149.

Consider a cube concentric with a parallelepiped
(rectangular box) with sides
$a<b<c$ and faces parallel
to that of the cube. Find the side length of the cube for which
the difference between the volume of the union and the volume of the
intersection of the cube and parallelepiped is minimum.

150.

The area of the bases of a truncated pyramid are equal
to
${S}_{1}$ and
${S}_{2}$ and the total area of the lateral surface is
$S$. Prove that, if there is a plane parallel to each of the bases
that partitions the truncated
pyramid into two truncated pyramids within
each of which a sphere can be inscribed, then
$S=(\sqrt{{S}_{1}}+\sqrt{{S}_{2}})(\sqrt[4]{{S}_{1}}+\sqrt[4]{{S}_{2}}){}^{2}\hspace{1em}.$