Solve the following system of equations:

$$
\frac{1}{2-x+2 y}-\frac{1}{x+2 y-1}=2,
$$

$$
\frac{1}{2-x+2 y}-\frac{1}{1-x-2 y}=4 .
$$
The first denominator is the same on the left side of both equations, and the second denominators can be transformed into each other by multiplication with (-1). Using this, we can simplify to a system of equations if we first consider the fractional expressions on the left side of the first equation as unknowns:

$$
\begin{aligned}
\frac{1}{2-x+2 y}=u & \text { and } & \frac{1}{x+2 y-1}=v \text{-with} \\
u-v=2, & & u+v=4
\end{aligned}
$$

From here, $u=3, v=1$. Now, from the

$$
2-x+2 y=\frac{1}{u}=\frac{1}{3}, \quad x+2 y-1=\frac{1}{v}=1
$$

system of equations, by adding, we get $4 y+1=4 / 3, y=1 / 12$, and by subtracting, we get $2 x-3=2 / 3, x=11 / 6$.

Emese Jambor (Budapest, Móricz Zs. Gymnasium, 2nd grade)