OMO Fall 2016

It is easy to see, that:

$log_{10}G = log_{10}10^{10^{100}} = 10^{100}$

Then, we can use the change of base theorem twice:

$log_{10^{100}}G = \frac {log_{10}G}{log_{10}10^{100}}= \frac {10^{100}}{100} = 10^{98}$

And finally:

$log_{10^{98}}G = \frac {log_{10}G}{log_{10}10^{98}} = \frac {10^{100}}{98} = \frac {5×10^{99}}{49}$

Now, it is also easy to see that:

$m= 5×10^{99} \text { and } m= 49 \text { (and that m and n are coprimes).}$

Therefore:

$m + n = 5×10^{99} + 49 = \overline {500...0049}$

(Our sum contains 99 - 2 = 97 zeroes in its decimal form.)

Hence, the sum of the digits:

$S = 5 + 97 × 0 + 4 + 9 = \boxed {18}$