2. (4p) a) Find the non-zero rational number $x$ that satisfies the equation:

$$
\frac{1}{x}+\frac{3}{x}+\frac{5}{x}+\ldots+\frac{2013}{x}=1+3+5+\ldots+105
$$

(3p) b) For $x$ determined above, solve the equation in the set of rational numbers

$$
(x-1): 10=y^{2}
$$

Doina Negrilă
2. a) $1+3+5+\ldots+2013=\frac{2014 \cdot 1007}{2}=1007^{2}$

$1+3+5+\ldots+105=\frac{106 \cdot 53}{2}=53^{2}$

The equation is equivalent to: $\frac{1007^{2}}{x}=53^{2} \Rightarrow x=\left(\frac{1007}{53}\right)^{2} \Rightarrow x=19^{2} \Rightarrow x=361$

b) $x=361 \Rightarrow(361-1): 10=y^{2} \Rightarrow y^{2}=36$

$S=\{-6,6\}$