Algebra for RMO

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Problem: Prove that 

\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}} is a rational number.

Proof:

Let x=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}. We then have

x - \sqrt[3]{2+\sqrt{5}} - \sqrt[3]{2-\sqrt{5}}=0

We know that a+b+c=0 implies that a^{3}+b^{3}+c^{3}=3abc, so we obtain

x^{3}-(2+\sqrt{5})-(2-\sqrt{5})=3x\sqrt[3]{(2+\sqrt{5})(2-\sqrt{5})}

or, x^{3}+3x-4=0

Clearly, out of the roots of this equation is x=1 and the other two roots satisfy the equation x^{2}+x+4=0 which has no real solutions. (This equation can be found by polynomial division). Since \sqrt[3]{2+\sqrt{5}}+\sqrt{2-\sqrt{5}} is a real number, it follows that

\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1, which is a rational number.

More later,

Nalin Pithwa

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