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# 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

${d}^{2}={a}^{2}+{b}^{2}+{c}^{2} .$

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 $1/\left(n+1\right)$ of the line was behind him. On Wednesday, the same fraction $x$ of the line was in front of him, while $1/\left(n+2\right)$ 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 $\left(n+1\right)/n$, where $n$ is a positive integer?

142.
Let $x,y>0$ be such that ${x}^{3}+{y}^{3}\le x-y$. 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 $\sqrt{\left(a+\mathrm{bx}\right)\left(c+\mathrm{dx}\right)}$ is rational. Explain the situation when $\mathrm{bc}=\mathrm{ad}$.