E.J. Barbeau
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.,
$V=\{x+y:x,y\in U,x\ne y\}\hspace{1em}.$
What is the smallest possible number of distinct elements that
$V$ can contain?

116.

Prove that the equation
${x}^{4}+5{x}^{3}+6{x}^{2}4x16=0$
has exactly two real solutions.

117.

Let
$a$ be a real number. Solve the
equation
$(a1)(\frac{1}{\mathrm{sin}x}+\frac{1}{\mathrm{cos}x}+\frac{1}{\mathrm{sin}x\mathrm{cos}x})=2\hspace{1em}.$

118.

Let
$a,b,c$ be nonnegative real numbers.
Prove that
${a}^{2}(b+ca)+{b}^{2}(c+ab)+{c}^{2}(a+bc)\le 3\mathrm{abc}\hspace{1em}.$
When does equality hold?

119.

The medians of a triangle
$\mathrm{ABC}$ intersect in
$G$.
Prove that
$\Vert \mathrm{AB}\Vert {}^{2}+\Vert \mathrm{BC}\Vert {}^{2}+\Vert \mathrm{CA}\Vert {}^{2}=3(\Vert \mathrm{GA}\Vert {}^{2}+\Vert \mathrm{GB}\Vert {}^{2}+\Vert \mathrm{GC}\Vert {}^{2})\hspace{1em}.$

120.

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