Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Find the answer

1. Given that the real number $x$ satisfies $20 \sin x=22 \cos x$ , then the greatest integer not exceeding the real number $\left(\frac{1}{\sin x \cos x}-1\right)^{7}$ is $\qquad$ .

Your answer

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

Solution

$\tan x=\frac{11}{10} \Rightarrow \sin 2 x=\frac{2 \tan x}{1+\tan ^{2} x}=\frac{\frac{11}{5}}{1+\frac{121}{100}}=\frac{220}{221} \text {, }$

Then $\left(\frac{1}{\sin x \cos x}-1\right)^{7}=\left(\frac{2}{\sin 2 x}-1\right)^{7}=\left(1+\frac{1}{110}\right)^{7} \in(1,2)$ , so the answer is 1.

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