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),...") |
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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: It is known that, for all integers , the following statement holds: The elements of are defined by the relation . Let Then, can be represented as a fraction , where and are relatively prime positive integers. Find .
Solution
From the given condition, we have:
Then