## Problem I - 5

A positive number $\mathrm{x}$ satisfies the relation

$$
x^{2}+\frac{1}{x^{2}}=7
$$

Prove that

$$
x^{5}+\frac{1}{x^{5}}
$$

is an integer and find its value.
Solution.

We have

$$
\left(x+\frac{1}{x}\right)^{2}=x^{2}+\frac{1}{x^{2}}+2=7+2=9 \Rightarrow x+\frac{1}{x}=3
$$

Then

$$
3 \cdot 9=27=\left(x+\frac{1}{x}\right)^{3}=x^{3}+\frac{1}{x^{3}}+3\left(x+\frac{1}{x}\right)=x^{3}+\frac{1}{x^{3}}+3 \cdot 3 \Rightarrow x^{3}+\frac{1}{x^{3}}=18
$$

Then

$$
7 \cdot 18=\left(x^{2}+\frac{1}{x^{2}}\right) \cdot\left(x^{3}+\frac{1}{x^{3}}\right)=x^{5}+\frac{1}{x^{5}}+x+\frac{1}{x}=3+x^{5}+\frac{1}{x^{5}} \Rightarrow x^{5}+\frac{1}{x^{5}}=123
$$