Digits in $2^n$

Let $a_n$ = number of odd digits in the base-$10$ expansion of $2^n$ and let $b_n$ = total number of digits in $2^n$.

For eg, $2^8 = 256$ and so $a_8 = 1$ ($5$ is the only odd digit) and $b_8 = 3$ (it is a 3 digit number).

Three problems:

A) Show that $$\sum_{n=1}^{\infty} \dfrac{a_n}{2^n} = \dfrac{1}{9}$$

B) Show that $$S = \sum_{n=1}^{\infty} \dfrac{b_n}{2^n}$$ is an irrational number.

C) Show that $S$ as defined in B) above satisfies $$ S \gt \dfrac{1169}{1023}$$


[Solution]

Sum of squares <= Sum of cubes

$a,b,c \gt 0$ and $abc = 1$

Show that

$$ a^2 + b^2 + c^2 \leq a^3 + b^3 + c^3$$

[Solution]