2022 AMC 10B Problem 5

Question

What is the value of

\frac{\left(1+\frac{1}{3}\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?

\sqrt3

2

\sqrt{15}

4

\sqrt{105}

Rewrite each square-root factor as a product of two nearby fractions

Pair each factor in the numerator with the matching factor under the square root

Let the given positive expression be

E.

Squaring it and using

1-\frac{1}{p^2}=(1-\frac{1}{p})(1+\frac{1}{p})

gives

\begin{aligned} E^2&=\prod_{p\in\{3,5,7\}}\frac{1+\frac{1}{p}}{1-\frac{1}{p}}\\ &=\frac42\cdot\frac64\cdot\frac86=4. \end{aligned}

Hence

E=2.

Thus, the answer is B .

Problem 5 in other years