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## PROBLEMS FOR OCTOBER

Valeria Pandelieva
641 Kirkwood Avenue
Ottawa, ON K1Z 5X5

109.
Suppose that

$\frac{{x}^{2}+{y}^{2}}{{x}^{2}-{y}^{2}}+\frac{{x}^{2}-{y}^{2}}{{x}^{2}+{y}^{2}}=k .$

Find, in terms of $k$, the value of the expression

$\frac{{x}^{8}+{y}^{8}}{{x}^{8}-{y}^{8}}+\frac{{x}^{8}-{y}^{8}}{{x}^{8}+{y}^{8}} .$

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 $\left\{T\right\}=\mathrm{AN}\cap \mathrm{CM}$, $\left\{R\right\}=\mathrm{BQ}\cap \mathrm{AN}$ and $\left\{S\right\}=\mathrm{CM}\cap \mathrm{BQ}$, 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) $f\left(4\right)=4$;
(d)

$\frac{1}{f\left(1\right)f\left(2\right)}+\frac{1}{f\left(2\right)f\left(3\right)}+\dots +\frac{1}{f\left(n\right)f\left(n+1\right)}=\frac{f\left(n\right)}{f\left(n+1\right)} .$

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.

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