Consider $f(x)=x\sqrt{8}+\frac{1}{x\sqrt{8}}-\sqrt{8}$. The function has 2 real roots, say, $x_1$ and $x_2$. If the $1994$th digit in the decimal expansion of $x_1$ is $6$, what is the $1994$th digit in the decimal expansion of $x_2$?
This is question 1 of the 34th Swedish Olympiad (1994). Either by explicitly finding the roots, or by Vieta's relations, we get that $x_1+x_2=1$. Now, if we write $1$ as $0.999\cdots$, we see that the corresponding digits in the decimal place add up to $9$. That is, the $m$th digit of $x_1$ plus the $m$th digit of $x_2=9$, for $m\geq2$. This means that the $1994$th digit of $x_2$ is $3$.
Is this correct? For some reason, I can't find the answer/solution anywhere.
Also, if we have $a,b\in\mathbb{R}$, and $(a+b)\in\mathbb{Z}$, then what can be said about the decimal expansion of $a$ and $b$? For example, $\pi=3.1415\cdots$, and $1-\pi=-2.1415\cdots$. We can see that the digits in the decimal expansion of $\pi$ and $1-\pi$ are equal (except for the first digit, of course). Can this be generalised?
Edit: I thought it would be helpful to include a screenshot that demonstrates that corresponding digits in the decimal expansions of $x_1$ and $x_2$ add up to $9$:
