AoPS - Problem

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Problem

Source: Indian TST Day 2 Problem 2

Tags: inequalities, triangle inequality, algebra unsolved, algebra

hajimbrak

11.07.2014 12:06

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$ .Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

ThirdTimeLucky

11.07.2014 17:29

Interpreting the

$v_j$ 's as the sides of a triangle due to statement 3 and the

$x_j$ 's and

$y_j$ 's as directed lengths created by three points, one point on each of the three sidelines of the triangle, statement 1 relates to the Menelaus theorem, statement 2 relates to the Carnot Theorem and statement 4 says that the triangle is acute.

trumpeter

05.01.2016 04:03

I have a similar problem here: Let $a_1$ , $a_2$ , and $a_3$ be positive reals such that $a_1+a_2+a_3>2\max\left(a_1,a_2,a_3\right)$ . Show that infinitely many $6$ -tuples $\left(x_1,x_2,x_3,x_4,x_5,x_6\right)$ of real numbers such that $a_1=x_1+x_4$ , $a_2=x_2+x_5$ , and $a_3=x_3+x_6$ satisfy $x_1x_2x_3=x_4x_5x_6$ and $x_1^2+x_2^2+x_3^2=x_4^2+x_5^2+x_6^2$ and given that $a_1$ , $a_2$ , and $a_3$ are not all the same, find $9$ distinct such $6$ -tuples.