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 $\ x\in (a;b).$

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 $\ x\in (a;b).$

Remark   there is something of a paradox due to the juxtaposition of the equation $\ s(x)=f(x)\ $ and the classical Jacques Hadamard equation $\ s(x)=\frac 12,\ $ which holds for almost all $\ x\in[0;1].$

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 solutions $\ x\in (a;b).$

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 $\ x\in (a;b).$