Maths Olympiad Prep

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, 2023

Algebra Difficulty 2.5 Junior Find the answer Canada

Suppose that a=1na = \dfrac{1}{n}, where nn is a
positive integer with n gt; 1\text{n gt; 1}.
Which of the following statements is true?

Pick one

Solution

Since a=1na = \dfrac{1}{n} where
nn is a positive integer with n gt; 1\text{n gt; 1}, then 0 lt; a lt; 1\text{0 lt; a lt; 1} and 1 a = n gt; 1\text{1 a = n gt; 1}.

Thus, 0 lt; a lt; 1 lt; 1 a\text{0 lt; a lt; 1 lt; 1 a}, which eliminates choices (D) and (E).

Since 0 lt; a lt; 1\text{0 lt; a lt; 1}, then a2a^2 is positive and a 2 lt; a\text{a 2 lt; a}, which eliminates choices (A)
and (C).

Thus, 0 lt; a 2 lt; a lt; 1 lt; 1 a\text{0 lt; a 2 lt; a lt; 1 lt; 1 a}, which tells us that (B) must be correct.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.