1. (4p) a) Determine the non-zero digits $a, b, c$ that simultaneously satisfy the conditions:
(i) $\overline{a b}+\overline{b a}=\overline{c c}$;
(ii) $\overline{a b}-\overline{b a}=c$;
(iii) $(\overline{c c}-c-c): c=c$.

(3p) b) Determine the number $\overline{a b}$, which satisfies the equation: $\overline{a b c d}-4 \cdot \overline{a b}-\overline{c d}=2016$.

Liviu Cocariu-Ardelean
1. a) From (iii) $(\overline{c c}-c-c): c=c \Rightarrow c=9$ ..... (1p)
From (i) $\overline{a b}+\overline{b a}=\overline{c c} \Rightarrow 11 a+11 b=99 \Rightarrow a+b=9$ ..... (1p)
From (ii) $\overline{a b}-\overline{b a}=c \Rightarrow 9 a-9 b=9 \Rightarrow a-b=1$ ..... (1p)
Finalization $a=5$ and $b=4$ ..... (1p)
b) $\overline{a b c d}-4 \cdot \overline{a b}-\overline{c d}=2016 \Leftrightarrow \overline{a b} \cdot 100+\overline{c d}-4 \cdot \overline{a b}-\overline{c d}=2016 \Leftrightarrow 96 \cdot \overline{a b}=2016$ ..... (2p)
Finalization $\overline{a b}=21$ ..... (1p)