Solve the following system of equations without using logarithms:

$$
a^{7 x} \cdot a^{15 y}=\sqrt{a^{19}}, \quad \sqrt[3]{a^{25 y}}: \sqrt{a^{13 x}}=\sqrt[12]{a}
$$
The first equation raised to the $2^{\text{nd}}$ power, the second equation raised to the $12^{\text{th}}$ power:

$$
a^{14 x+30 y}=a^{19}, a^{100 y-78}=a
$$

or

$$
14 x+30 y=19, \quad 100 y-78 x=1
$$

from which

$$
x=\frac{1}{2}, \quad y=\frac{2}{5}
$$

(Zoltán Reich, Zenta.)

The problem was also solved by: J. Felhőssy, P. Füstös, Z. Harsányi, P. Heimlich, Gy. Jánosy, J. Kiss, E. Makó, S. Pichler, E. Sárközy, Gy. Schlesinger, Gy. Schuster, J. Szabó, J. Steiger, B. Tóth.