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# PROBLEMS FOR MAY

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 $\angle A={90}^{ˆ}$. 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

$\frac{{x}_{1}{x}_{2}\dots {x}_{n}}{\left(1+{x}_{1}\right)\left({x}_{1}+{x}_{2}\right)\dots \left({x}_{n-1}+{x}_{n}\right)\left({x}_{n}+{a}^{n+1}\right)}$

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) $\left\{{a}_{0},{a}_{1},\dots ,{a}_{n}\dots \right\}$ satisfying, for each positive integer $n$, the equation

$\frac{{a}_{0}}{{a}_{1}}+\frac{{a}_{0}}{{a}_{2}}+\dots +\frac{{a}_{0}}{{a}_{n}}+\frac{p}{{a}_{n+1}}=1 .$

149.
Consider a cube concentric with a parallelepiped (rectangular box) with sides $a 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

$S=\left(\sqrt{{S}_{1}}+\sqrt{{S}_{2}}\right)\left(\sqrt[4]{{S}_{1}}+\sqrt[4]{{S}_{2}}\right){}^{2} .$