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Problem 726

AMC 10/12, early questions

Algebra Difficulty 4.0 Multiple choice Progetto Olimpiadi di Matematica GARA di SECONDO LIVELLO BIENNIO · Italy · 2003

Given two numbers $x$ and $y$ with $0 < x < 1$ , $0 < y < 1$ , which of the following statements is always true?

Pick one

Official solution

Solution:

The answer is $\mathbf{(E)}$ . This statement is always true, since $x < 1$ , while $\frac{1}{y} > 1$ . For $x = \frac{1}{2}$ , $y = \frac{1}{3}$ the hypotheses of the text are satisfied, but (B), (C) and (D) are false. Finally $x = y = \frac{3}{4}$ provides a counterexample for (A).

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.