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Calculus Level 3

f(x)=m=0  n=0mxn  n=0xn\large f(x)=\prod _{ m=0 }^{ \infty }{\frac {\ \ \displaystyle \sum _{ n=0 }^{ m }{ { x }^{ n }\ \ } }{\displaystyle \sum _{ n=0 }^{ \infty }{ { x }^{ n } } } }

Given f(x)f(x) defined as above, define C=2120×f(12).C=\left\lfloor { 2 }^{ 120 } \times f\left(\frac { 1 }{ 2 } \right) \right\rfloor. Then, how many 1's does the binary representation of CC contain? That is, what is the binary digit sum of C?C?


Hint: You do not need a computer to solve this problem, nor do you need to calculate f(12)f\left(\frac{1}{2}\right) to a high accuracy. There is a more "contest-friendly" way to solve this problem.


Bonus: Can you give a formula for the digit sum of B120×f(1B)\left\lfloor { B }^{ 120 } \times f\left(\frac { 1 }{ B } \right) \right\rfloor in base B?B?

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