Difference between revisions of "2023 SSMO Team Round Problems/Problem 13"
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==Problem== | ==Problem== | ||
Let <math>D(n)</math> denote the product of all divisors of <math>n</math> Let <math>P(i,j)</math> denote the set of all integers that are both a multiple of <math>i</math> and a factor of <math>j.</math> Let | Let <math>D(n)</math> denote the product of all divisors of <math>n</math> Let <math>P(i,j)</math> denote the set of all integers that are both a multiple of <math>i</math> and a factor of <math>j.</math> Let | ||
| − | + | <cmath> | |
-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)}}. | -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)}}. | ||
| − | + | </cmath> | |
Suppose <math>\sum_{k=2}^{\infty}G(k)</math> is <math>\frac{a+b\sqrt{c}}{d}</math>. Find the value of <math>a+b+c+d</math>. | Suppose <math>\sum_{k=2}^{\infty}G(k)</math> is <math>\frac{a+b\sqrt{c}}{d}</math>. Find the value of <math>a+b+c+d</math>. | ||
==Solution== | ==Solution== | ||
Latest revision as of 19:26, 2 May 2025
Problem
Let 
denote the product of all divisors of 
Let 
denote the set of all integers that are both a multiple of 
and a factor of 
Let
![\[-F(a) = \sqrt{\left|\log_{10}\left(\frac{D(10^{a})}{\prod_{\omega\in P(10^2,10^{a+2})}\omega}\right)\right|}\text{ and }G(n) = \sqrt[n-1]{\prod_{i=2}^{n}10^{F(i)}}.\]](http://latex.artofproblemsolving.com/9/0/d/90d8e6c0eb483f20808b887b2b6f2d9885535067.png)
Suppose 
is 
. Find the value of 
.
