## Subject II. (20 points)

a) Determine the natural numbers $a$ and $b$ knowing that $b$ is prime and that $a^{2}+a^{4}+b^{6}=20944$;

b) Determine the numbers $x, y, z$ knowing that $x+y, y+z, z+x$ are directly proportional to $5 ; 9 ; 12$ and that $\frac{5 x y-2 y z}{z+y-x}=36$.

Prof. Vasile Şerdean, Şcoala Gimnazială nr. 1 Gherla
Subject II.
a) $a^{2} \cdot\left(a^{2}+1\right)+b^{6}=20944$

$$
a^{2} \cdot\left(a^{2}+1\right) \text { and } 20944 \text { are even } \Rightarrow b \text { is prime and even } \Rightarrow b=2 \quad \text { (5 points) }
$$

$$
\begin{aligned}
& \Rightarrow a^{2} \cdot\left(a^{2}+1\right)=20944-2^{6} \Leftrightarrow a^{2} \cdot\left(a^{2}+1\right)=20880 \Leftrightarrow \\
& a^{2} \cdot\left(a^{2}+1\right)=12^{2} \cdot\left(12^{2}+1\right) \Leftrightarrow a=12
\end{aligned}
$$

(5 points)
b) $x+y+z=13 k \Rightarrow\left\{\begin{array}{l}x=4 k \\ y=k \\ z=8 k\end{array} \Rightarrow k=45 \Rightarrow\left\{\begin{array}{l}x=180 \\ y=45 \\ z=360\end{array}\right.\right.$

(10 points)