MathDB

3

Part of 2012 European Mathematical Cup

Problems(2)

Non-linear System of Equations with 3 Variables

Source: European Mathematical Cup 2012, Junior Division, Problem 3

7/27/2013
Are there positive real numbers xx, yy and zz such that
x4+y4+z4=13, x^4 + y^4 + z^4 = 13\text{,}
x3y3z+y3z3x+z3x3y=63, x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,}
x3yz+y3zx+z3xy=53? x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?}
Proposed by Matko Ljulj.
algebrapolynomialVietasystem of equationsalgebra proposed
Inequality in 6 Variables.

Source: European Mathematical Cup 2012, Senior Division, Problem 3

7/27/2013
Prove that the following inequality holds for all positive real numbers aa, bb, cc, dd, ee and ff
abca+b+d3+defc+e+f3<(a+b+d)(c+e+f)3.\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}
Proposed by Dimitar Trenevski.
inequalitiesinequalities proposed