A hard VMO Problem #1

Algebra Level 3

Consider the graph of the function y = x 2 3 y=\sqrt[3]{x^{2}} .
For any straight line d d that intersects ( C ) (C) at 3 points, let the x x -coordinates of the intersection points be x 1 , x 2 , x_{1}, x_{2}, and x 3 x_{3} respectively,

Is the value of A = x 1 x 2 x 3 2 3 + x 2 x 3 x 1 2 3 + x 3 x 1 x 2 2 3 A=\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}} always constant?

Bonus: Can you prove the correct answer?

Yes No A A doesn't give any valid values.

Tin Le by Tin Le , 15, Vietnam

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1 solution

Chris Lewis
Jun 16, 2019

Say the line has equation y = m x + c y=mx+c . From the equations of the curve, we have y 3 = x 2 y^3=x^2 . So at the intersection points, we have

x 2 = m 3 x 3 + 3 c m 2 x 2 + 3 c 2 m x + c 3 x^2=m^3 x^3 + 3cm^2 x^2 + 3c^2 mx + c^3

or

m 3 x 3 + ( 3 c m 2 − 1 ) x 2 + 3 c 2 m x + c 3 = 0 m^3 x^3 + \left( 3cm^2 -1 \right) x^2 + 3c^2 mx + c^3 = 0

The quantity we want to find is

A = x 2 x 3 x 1 2 3 + x 3 x 1 x 2 2 3 + x 1 x 2 x 3 2 3 = x 1 x 2 x 3 3 ( 1 x 1 + 1 x 2 + 1 x 3 ) = ( x 1 x 2 x 3 ) − 2 3 ( x 2 x 3 + x 3 x 1 + x 1 x 2 ) A=\sqrt[3]{\frac{x_2 x_3}{x_1^2}}+\sqrt[3]{\frac{x_3 x_1}{x_2^2}}+\sqrt[3]{\frac{x_1 x_2}{x_3^2}}=\sqrt[3]{x_1 x_2 x_3} \left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}\right)=\left(x_1 x_2 x_3\right)^{\frac{-2}{3}}(x_2 x_3 + x_3 x_1 + x_1 x_2)

By Vieta, x 1 x 2 x 3 = c 3 m 3 x_1 x_2 x_3=\frac{c^3}{m^3} and x 2 x 3 + x 3 x 1 + x 1 x 2 = 3 c 2 m 2 x_2 x_3 + x_3 x_1 + x_1 x_2=\frac{3c^2}{m^2} .

Substituting in, we find A = 3 A=\boxed3 , ie it is constant.

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