Sunday, May 31, 2015

China IMO Mock Problem: Given $xy$ is rational number, and $x$ and $y$ are the roots of the equations $6x^2+2015x+8=0$ and $6x^2+2015x+8=0$ respectively. Evaluate $\dfrac{x}{y}$.

Given $xy\ne 1$, and $x$ and $y$ are the roots of the equations $6x^2+2015x+8=0$ and $6x^2+2015x+8=0$ respectively. Evaluate $\dfrac{x}{y}$.


Since the coefficients of $6x^2+2015x+8=0$ are the same as the coefficients of $8y^2+2015y+6$ in reversed order, and we're told that $x$ is the root of the equation $6x^2+2015x+8=0$ while $y$ is the root of the equation  $8y^2+2015y+6$.

Therefore, we have $x,\,\dfrac{1}{y}$ are the roots of the equation $6x^2+2015x+8=0$ and hence their product is $\dfrac{8}{6}=\dfrac{4}{3}$.

Suppose we have not realized the reversed order of the coefficients between the two given quadratic equations, we can still manage this problem though the tedious way out, i.e. to solve for both equations for the values of $x$ and $y$ and then find for their product, and bear in mind that $xy$ is a rational number:

Solving $6x^2+2015x+8=0$ for $x$, we get:

$\begin{align*}x&=\dfrac{-2015\pm \sqrt{2015^2-4(6)(8)}}{2(6)}\\&=\dfrac{-2015\pm \sqrt{4,060,033}}{2(6)}\end{align*}$

Solving $8y^2+2015y+6$ for $y$ we get:

$\begin{align*}y&=\dfrac{-2015\pm \sqrt{2015^2-4(6)(8)}}{2(8)}\\&=\dfrac{-2015\pm \sqrt{4,060,033}}{2(8)}\end{align*}$

Since we're told  $xy$ is a rational number, we need to choose wisely for the pair of $x$ and $y$:

If we pick $x=\dfrac{-2015+\sqrt{4,060,033}}{2(6)}$ then $y$ has to be $\dfrac{-2015+\sqrt{4,060,033}}{2(8)}$ and multiply them together, we get:

$\begin{align*}\dfrac{x}{y}&=\dfrac{\dfrac{-2015+\sqrt{4,060,033}}{2(6)}}{\dfrac{-2015+\sqrt{4,060,033}}{2(8)}}\\&=\dfrac{-2015+\sqrt{4,060,033}}{2(6)}\times \dfrac{2(8)}{-2015+\sqrt{4,060,033}}\\&=\dfrac{\cancel{-2015+\sqrt{4,060,033}}}{\cancel{2}(6)}\times \dfrac{\cancel{2}(8)}{\cancel{-2015+\sqrt{4,060,033}}}\\&=\dfrac{4}{3}\end{align*}$

and we're done.

Although the second method works, we need to aware that as a 21st century learners, we have to train ourselves to become independent, excellent and QUICK problem solver! We should be passionate about learning new things and to be a critical thinker.

The observation in the first heuristic problem solving method save us great time in tackling this problem successfully and elegantly. The good news is, like I have always said, we can learn the heuristic problem solving skills from others.

Note that this is a problem mentioned here as well:

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