## Task 10/74

The task is to find all real numbers $x$ for which the following equation holds:

$$
\sin (\pi \lg x)+\cos (\pi \lg x)=1
$$
We set $\pi \cdot \lg x=\alpha$. Then the equation takes the form $\sin \alpha+\cos \alpha=1$. The solutions of this trigonometric equation are $\alpha_{1}=2 k \pi$ and $\alpha_{2}=\left(2 k+\frac{1}{2}\right) \pi$.

(where $k$ runs through the set of all integers). From this it follows that

$$
\lg x_{1}=2 k ; \lg x_{2}=2 k+\frac{1}{2} \quad \text { so } \quad x_{1}=10^{2 k}=100^{k} ; x_{2}=10^{2 k+0.5}=100^{k} \sqrt{10}
$$