2023 SSMO Team Round Problems/Problem 13

Problem

Let $D(n)$ denote the product of all divisors of $n$ Let $P(i,j)$ denote the set of all integers that are both a multiple of $i$ and a factor of $j.$ Let $-F(a) = \sqrt{\left|\log_{10}\left(\frac{D(10^{a})}{\prod_{\omega\in P(10^2,10^{a+2})}\omega}\right)\right|}\text{ and }G(n) = \sqrt[n-1]{\prod_{i=2}^{n}10^{F(i)}}.$ Suppose $\sum_{k=2}^{\infty}G(k)$ is $\frac{a+b\sqrt{c}}{d}$ . Find the value of $a+b+c+d$ .

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2023 SSMO Team Round Problems/Problem 13

Problem

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