1. a) Given that $x^{2}+y^{2}-8 x \sqrt{3}-6 y \sqrt{2}+66=0, x, y \in R$, compare the numbers $z$ and $t$, where $z=\frac{[x]-[y]}{[-x]-[-y]}$ and $t=\frac{1+\sqrt{2}}{2}$

b) Given that $x+\frac{1}{x}=10$, calculate the value of the expression $E(x)=x+x^{2}+x^{3}+x^{4}+\frac{1}{x^{4}}+\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{1}{x}$
1. a) Given that $x^{2}+y^{2}-8 x \sqrt{3}-6 y \sqrt{2}+66=0$, x, y $\in \mathrm{R}$, compare the numbers $z$ and $\mathrm{t}$, where $z=\frac{[x]-[y]}{[-x]-[-y]}$ and $t=\frac{1+\sqrt{2}}{2}$

(3p)

## SOLUTION

$(x-4 \sqrt{3})^{2}+(y-3 \sqrt{2})^{2}=0 \Rightarrow x=4 \sqrt{3}$ and $y=3 \sqrt{2}$

(2p)

$z=\frac{[4 \sqrt{3}]-[3 \sqrt{2}]}{[-4 \sqrt{3}]-[-3 \sqrt{2}]} \Rightarrow z=\frac{6-4}{-7-(-5)}=\frac{2}{-2}=-1 \Rightarrow z<t$

(1p)

b) Given that $x+\frac{1}{x}=10$, calculate the value of the expression $E(x)=x+x^{2}+x^{3}+x^{4}+\frac{1}{x^{4}}+\frac{1}{x^{3}}+\frac{1}{x^{2}}+\frac{1}{x}$ (4p)

## SOLUTION

We successively obtain: $x^{2}+\frac{1}{x^{2}}=98 ; x^{3}+\frac{1}{x^{3}}=970 ; x^{4}+\frac{1}{x^{4}}=9602$

(3p)

$\mathrm{E}(\mathrm{x})=10+98+970+9602=10680$

(1p)