2021 JMC 10 Problems/Problem 2

Problem

There exist irrational numbers $e\approx 2.72$ and $\pi \approx 3.14.$ How can $|\pi - |e - | e - \pi|||$ be expressed in terms of $\pi$ and $e?$

$\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e$

Solution

Note that $e< \pi$ and $2e >\pi.$ So, it follows that $|\pi - | e - |e-\pi||| = |\pi - | e - (\pi - e)|| = |\pi - |2e - \pi|| = |2\pi - 2e| = 2\pi - 2e.$

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2021 JMC 10 Problems/Problem 2

Problem

Solution