Olympiad Maths Prep

Track / Stage 4 / 327 of 340 #587 of 2000

Problem 587

AMC 12 late, AIME early
Algebra Difficulty 5.0 Find the answer

I3.2 Given that x+1x=Px+\frac{1}{x}=P. If x6+1x6=Qx^{6}+\frac{1}{x^{6}}=Q, find the value of QQ.

Official solution

x+1x=1(x+1x)2=1⇒x2+1x2=−1(x2+1x2)3=−1⇒x6+1x6+3(x2+1x2)=−1⇒x6+1x6=2∴Q=2\begin{array}{l}x+\frac{1}{x}=1 \\ \left(x+\frac{1}{x}\right)^{2}=1 \\ \Rightarrow x^{2}+\frac{1}{x^{2}}=-1 \\ \left(x^{2}+\frac{1}{x^{2}}\right)^{3}=-1 \\ \Rightarrow x^{6}+\frac{1}{x^{6}}+3\left(x^{2}+\frac{1}{x^{2}}\right)=-1 \\ \Rightarrow x^{6}+\frac{1}{x^{6}}=2 \\ \therefore Q=2\end{array}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.