Problem 1. The angle $\alpha$ is greater than its complementary angle by as many times as it is smaller than its supplementary angle. Calculate the angle $\alpha$.

翻译完成，但请注意，原文本中的数学问题已经用英文表述。如果您需要解题帮助，请告知。
Solution. According to the condition of the problem, there exists a number $k$ such that

$$
\alpha=k\left(90^{\circ}-\alpha\right)
$$

$$
180^{\circ}-\alpha=k \alpha
$$

Then,

$$
\begin{aligned}
& k \alpha+\alpha=180^{\circ} \\
& k \alpha+\alpha=k \cdot 90^{\circ}
\end{aligned}
$$

According to this, $k \cdot 90^{\circ}=180^{\circ}$, from which we have $k=2$. Then, $3 \alpha=180^{\circ}$, i.e., $\alpha=180^{\circ}: 3=60^{\circ}$.