Maths Olympiad Prep

Library / /99 of 860

Algebra Difficulty 4.8 AIME Find the answer

Reduce the number 2+53+2−53\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Observe that (2+53+2−53)3=(2+5)−3(2+53)−3(2−53)+(2−5)=4−3(2+53+2−53)(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}})^{3}=(2+\sqrt{5})-3(\sqrt[3]{2+\sqrt{5}})-3(\sqrt[3]{2-\sqrt{5}})+(2-\sqrt{5})=4-3(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}) Hence 2+53+2−53\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}} is a root of the cubic x3+3x−4=(x−1)(x2+x+4)x^{3}+3 x-4=(x-1)(x^{2}+x+4). The roots of x2+x+4x^{2}+x+4 are imaginary, so 2+53+2−53=1\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=\mathbf{1}.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.