MathDB

5

Part of 2004 Federal Competition For Advanced Students, P2

Problems(1)

4x4 system, symmetry, a^2 = (\sqrt{bc}\sqrt[3]{bcd})/(b+c)(b+c+d)

Source: Austrian Federal Competition For Advanced Students 2004, Part 2, p5

8/30/2019
Solve the following system of equations in real numbers: {a2=bcbcd3(b+c)(b+c+d)b2=cdcda3(c+d)(c+d+a)c2=dadab3(d+a)(d+a+b)d2=ababc3(a+b)(a+b+c)\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\ b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\ c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\ d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}
algebrasystem of equations