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# Problems

E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than December 31, 2001. Please make sure that your name, email address and complete mailing address are on the first page.

115.
Let $U$ be a set of $n$ distinct real numbers and let $V$ be the set of all sums of distinct pairs of them, i.e.,

$V=\left\{x+y:x,y\in U,x\ne y\right\} .$

What is the smallest possible number of distinct elements that $V$ can contain?

116.
Prove that the equation

${x}^{4}+5{x}^{3}+6{x}^{2}-4x-16=0$

has exactly two real solutions.

117.
Let $a$ be a real number. Solve the equation

$\left(a-1\right)\left(\frac{1}{\mathrm{sin}x}+\frac{1}{\mathrm{cos}x}+\frac{1}{\mathrm{sin}x\mathrm{cos}x}\right)=2 .$

118.
Let $a,b,c$ be nonnegative real numbers. Prove that

${a}^{2}\left(b+c-a\right)+{b}^{2}\left(c+a-b\right)+{c}^{2}\left(a+b-c\right)\le 3\mathrm{abc} .$

When does equality hold?

119.
The medians of a triangle $\mathrm{ABC}$ intersect in $G$. Prove that

$‖\mathrm{AB}‖{}^{2}+‖\mathrm{BC}‖{}^{2}+‖\mathrm{CA}‖{}^{2}=3\left(‖\mathrm{GA}‖{}^{2}+‖\mathrm{GB}‖{}^{2}+‖\mathrm{GC}‖{}^{2}\right) .$

120.
Determine all pairs of nonnull vectors x, y for which the following sequence $\left\{{a}_{n}:n=1,2,\dots \right\}$ is (a) increasing, (b) decreasing, where

${a}_{n}=‖x-ny‖ .$