PROBLEMS FOR AUGUST
Please send your solutions to
E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later that September 30, 2001. Unless you are
submitting in TeX, please do not submit your solutions as
an electronic attachment. Make sure that your
name, complete mailing address and email address are on the
front page of your solution set.
Notes. A rectangular hyperbola is an hyperbola
whose asymmptotes are at right angles.

97.

A triangle has its three vertices on a rectangular
hyperbola. Prove that its orthocentre also lies on the
hyperbola.

98.

Let
${a}_{1},{a}_{2},\dots ,{a}_{n+1},{b}_{1},{b}_{2},\dots ,{b}_{n}$ be nonnegative real numbers for which
(i)
${a}_{1}\ge {a}_{2}\ge \dots \ge {a}_{n+1}=0$,
(ii)
$0\le {b}_{k}\le 1$ for
$k=1,2,\dots ,n$.


Suppose that
$m=\lfloor {b}_{1}+{b}_{2}+\dots +{b}_{n}\rfloor +1$. Prove that
$\sum _{k=1}^{n}{a}_{k}{b}_{k}\le \sum _{k=1}^{m}{a}_{k}\hspace{1em}.$

99.

Let
$E$ and
$F$ be respective points on sides
$\mathrm{AB}$ and
$\mathrm{BC}$ of a triangle
$\mathrm{ABC}$ for which
$\mathrm{AE}=\mathrm{CF}$. The
circle passing through the points
$B,C,E$ and the circle
passing through the points
$A,B,F$ intersect at
$B$ and
$D$. Prove that
$\mathrm{BD}$ is the bisector of angle
$\mathrm{ABC}$.

100.

If 10 equally spaced points around a circle
are joined consecutively, a convex regular inscribed decagon
$P$ is obtained; if every third point is joined, a
selfintersecting regular decagon
$Q$ is formed. Prove that
the difference between the length of a side of
$Q$ and
the length of a side of
$P$ is equal to the radius of the
circle. [With thanks to Ross Honsberger.]

101.

Let
$a,b,u,v$ be nonnegative. Suppose that
${a}^{5}+{b}^{5}\le 1$ and
${u}^{5}+{v}^{5}\le 1$. Prove that
${a}^{2}{u}^{3}+{b}^{2}{v}^{3}\le 1\hspace{1em}.$
[With thanks to Ross
Honsberger.]

102.

Prove that there exists a tetrahedron
$\mathrm{ABCD}$, all
of whose faces are similar right triangles, each face having
acute angles at
$A$ and
$B$. Determine which of the edges of
the tetrahedron is largest and which is smallest, and find the
ratio of their lengths.