Ágnes added the last two digits of her birth year, then added their product, and noticed that she is exactly that age this year. When was Ágnes born?
Let the last two digits of Ágnes's birth year be $x$ and $y$. Then we can write the following equation:

$$
x+y+x y=92-(10 x+y)
$$

Rearranging:

$$
\begin{gathered}
11 x+2 y+x y=92 \\
(x+2) \cdot(y+11)=114
\end{gathered}
$$

Since $x, y$ are integers, $0 \leq x, y \leq 9$ and $(11+y) \mid 114=2 \cdot 3 \cdot 19$, thus $11+y=19$ is the only possibility. Then $y=8, x=4$; hence, Ágnes was born in 1948.