
109.

Suppose that
$\frac{{x}^{2}+{y}^{2}}{{x}^{2}{y}^{2}}+\frac{{x}^{2}{y}^{2}}{{x}^{2}+{y}^{2}}=k\hspace{1em}.$
Find, in terms of
$k$, the value of the expression
$\frac{{x}^{8}+{y}^{8}}{{x}^{8}{y}^{8}}+\frac{{x}^{8}{y}^{8}}{{x}^{8}+{y}^{8}}\hspace{1em}.$

110.

Given a triangle
$\mathrm{ABC}$ with an area of 1.
Let
$n>1$ be a natural number. Suppose that
$M$ is a point on the side
$\mathrm{AB}$ with
$\mathrm{AB}=\mathrm{nAM}$,
$N$ is a point on the side
$\mathrm{BC}$ with
$\mathrm{BC}=\mathrm{nBN}$, and
$Q$ is a point on the side
$\mathrm{CA}$ with
$\mathrm{CA}=\mathrm{nCQ}$.
Suppose also that
$\{T\}=\mathrm{AN}\cap \mathrm{CM}$,
$\{R\}=\mathrm{BQ}\cap \mathrm{AN}$ and
$\{S\}=\mathrm{CM}\cap \mathrm{BQ}$, where
$\cap $ signifies that the singleton
is the intersection of the indicated segments. Find the
area of the triangle
$\mathrm{TRS}$ in terms of
$n$.

111.

(a) Are there four different numbers, not exceeding
10, for which the sum of any three is a prime number?


(b) Are there five different natural numbers such
that the sum of every three of them is a prime number?

112.

Suppose that the measure of angle
$\mathrm{BAC}$ in the
triangle
$\mathrm{ABC}$ is equal to
$\alpha $. A line passing through
the vertex
$A$ is perpendicular to the angle bisector of
$\angle \mathrm{BAC}$ and intersects the line
$\mathrm{BC}$ at the point
$M$.
Find the other two angles of the triangle
$\mathrm{ABC}$ in terms of
$\alpha $, if it is known that
$\mathrm{BM}=\mathrm{BA}+\mathrm{AC}$.

113.

Find a function that satisfies all of the following
conditions:


(a)
$f$ is defined for every positive integer
$n$;


(b)
$f$ takes only positive values;


(c)
$f(4)=4$;


(d)
$\frac{1}{f(1)f(2)}+\frac{1}{f(2)f(3)}+\dots +\frac{1}{f(n)f(n+1)}=\frac{f(n)}{f(n+1)}\hspace{1em}.$

114.

A natural number is a multiple of 17. Its binary
representation (i.e., when written to base 2) contains
exactly three digits equal to 1 and some zeros.


(a) Prove that there are at least six digits equal
to 0 in its binary representation.


(b) Prove that, if there are exactly seven digits equal
to 0 and three digits equal to 1, then the number must be even.