2

Part of 2009 Hungary-Israel Binational

Problems(2)

roots of a cubic

Source: Hungary-Israel Binational Olympiad 2009, Problem 2

Denote the three real roots of the cubic x^3 \minus{} 3x \minus{} 1 \equal{} 0 by

x_1

x_2

x_3

in order of increasing magnitude. (You may assume that the equation in fact has three distinct real roots.) Prove that x_3^2 \minus{} x_2^2 \equal{} x_3 \minus{} x_1.

algebra unsolvedalgebra

Inequality 3 variables - seems familiar

Source: Hungary-Israel Binational Olympiad 2009, Problem 5

x

y

z

be non negative numbers. Prove that \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}

inequalitiesquadratics