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Difference between revisions of "Northeastern WOOTers Mock AIME I Problems/Problem 14"

(Created page with "== Problem 14 == Consider three infinite sequences of real numbers: <cmath> \begin{eqnarray*} X &=& \left( x_1, x_2, \cdots \right), \\ Y &=& \left( y_1, y_2, \cdots \right),...")
 
m (→‎Solution)
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<cmath>\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 \\
 
<cmath>\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*} </cmath>
+
\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) \\
 +
\left( \left( \log_2 y_n \right)^2 - \log_2 x_n \log_2 z_n \right)^2 &= 0 \\
 +
\left( \log_2 y_n \right)^2 &= \log_2 x_n \log_2 z_n \end{align*} </cmath>
 +
 
 +
Then

Revision as of 17:05, 8 August 2021

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) \\ \left( \left( \log_2 y_n \right)^2 - \log_2 x_n \log_2 z_n \right)^2 &= 0 \\ \left( \log_2 y_n \right)^2 &= \log_2 x_n \log_2 z_n \end{align*}

Then

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