Maths Olympiad Prep

Library / /252 of 860

Algebra Difficulty 5.0 AIME, harder Find the answer

The equation x2+2x=ix^{2}+2 x=i has two complex solutions. Determine the product of their real parts.

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

Solution

The solutions are x+1=±eiπ824x+1= \pm e^{\frac{i \pi}{8}} \sqrt[4]{2}. The desired product is then (−1+cos⁡(π8)24)(−1−cos⁡(π8)24)=1−cos⁡2(π8)2=1−(1+cos⁡(π4))22=1−22\left(-1+\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)\left(-1-\cos \left(\frac{\pi}{8}\right) \sqrt[4]{2}\right)=1-\cos ^{2}\left(\frac{\pi}{8}\right) \sqrt{2}=1-\frac{\left(1+\cos \left(\frac{\pi}{4}\right)\right)}{2} \sqrt{2}=\frac{1-\sqrt{2}}{2}

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.