Home Competitions ÖMO 2004 Problem 5
Solve the following system of equations in real numbers: { a 2 = b c b c d 3 ( b + c ) ( b + c + d ) b 2 = c d c d a 3 ( c + d ) ( c + d + a ) c 2 = d a d a b 3 ( d + a ) ( d + a + b ) d 2 = a b a b c 3 ( 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} { a 2 = ( b + c ) ( b + c + d ) b c 3 b c d b 2 = ( c + d ) ( c + d + a ) c d 3 c d a c 2 = ( d + a ) ( d + a + b ) d a 3 d ab d 2 = ( a + b ) ( a + b + c ) ab 3 ab c
Dataset text AI-MO AoPS . Quoted with source.
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