2014 AIME I Problems/Problem 9

Problem 9

Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$ . Find $x_2(x_1+x_3)$ .

Solution

Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{2014} x^3 - 4029x^2 + 2 = 0$ . Find $x_2(x_1+x_3)$ .

We note that $x=\dfrac{1}{\sqrt{2014}}$ is a solution since $(\dfrac{1}{\sqrt{2014}})^3*\sqrt{2014}-4029(\dfrac{1}{\sqrt{2014}})^2+2=\dfrac{1}{2014}-\dfrac{4029}{2014}+2=\dfrac{1-4029+4028}{2014} = 0$

We claim that $x_2=\dfrac{1}{\sqrt{2014}}$

by vieta's formula we have that the $x^2$ coefficent is equal to $-x_1-x_2-x_3$ and that the $x$ coeefficent is equal to $x_1x_2+x_1x_3+x_2x_3$ so using the values in the above equation we get: $-x_1-x_2-x_3=-4029\rightarrow x_1+x_2+x_3=4029$

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2014 AIME I Problems/Problem 9

Problem 9

Solution