Northeastern WOOTers Mock AIME I Problems/Problem 14

Problem 14

Consider three infinite sequences of real numbers: $\begin{eqnarray*} X &=& \left( x_1, x_2, \cdots \right), \\ Y &=& \left( y_1, y_2, \cdots \right), \\ Z &=& \left( z_1, z_2, \cdots \right). \end{eqnarray*}$ It is known that, for all integers $n$ , the following statement holds: $\begin{align*} \left( \left( \log_2 x_n \right)^2 + \left( \log_2 y_n \right)^2 \right) \cdot \left( \left( \log_2 y_n \right)^2 + \left( \log_2 z_n \right)^2 \right) \\ &= \left( \log_2 x_n \log_2 y_n + \log_2 y_n \log_2 z_n \right)^2.\end{align*}$ The elements of $Y$ are defined by the relation $y_n=2^{\frac{n}{2^n}}$ . Let $S =\sum_{n=1}^{\infty} \log_2 x_n \log_2 y_n \log_2 z_n.$ Then, $S$ can be represented as a fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$ .

Solution

From the given condition, we have:

$\begin{align*} \left( \left( \log_2 x_n \right)^2 + \left( \log_2 y_n \right)^2 \right) \cdot \left( \left( \log_2 y_n \right)^2 + \left( \log_2 z_n \right)^2 \right) &= \left( \log_2 x_n \log_2 y_n + \log_2 y_n \log_2 z_n \right)^2 \\ \left( \log_2 y_n \right)^4 + \left( \log_2 x_n \log_2 z_n \right)^2 &= 2 \left( \log_2 y_n \right)^2 \left( \log_2 x_n \log_2 z_n \right) \end{align*}$

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Northeastern WOOTers Mock AIME I Problems/Problem 14

Problem 14

Solution