The following open question was popular in some places:
Q: does there exist $\ x\in(0;1)\ $ such that $\,\ s(x)=x,\,\ $ where
$$ s(x)\ :=\ \lim_{n=\infty} \frac{\sum_{k=1}^n x_k}n $$
and $\ x_n\ $ are binary digits $\ 0\ $ or $\ 1,\ $ and
$$ x\ =\ \sum_{n=1}^\infty\frac{x_n}{2^n} $$
Thus, in 1959/60 I've formulated and proved a more general theorem, and it was recognized in the spring of 1961, namely:
Theorem For every function $\ f: (0;1)\rightarrow (0;1)\ $ which is a pointwise limit of a sequence of continuous functions, and for every non-empty $\ (a;b)\subseteq (0;1),\ $ the equation
$$ s(x)\ =\ f(x) $$
has $\ 2^{\aleph_0} $ of different solutions $\ x\in (a;b).$