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### PROBLEMS FOR AUGUST

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 $m=⌊{b}_{1}+{b}_{2}+\dots +{b}_{n}⌋+1$. Prove that

$\sum _{k=1}^{n}{a}_{k}{b}_{k}\le \sum _{k=1}^{m}{a}_{k} .$

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 self-intersecting 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

${a}^{2}{u}^{3}+{b}^{2}{v}^{3}\le 1 .$

[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.
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