Maths Olympiad Prep

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Number theory Difficulty 6.1 National olympiad Prove it

9. In the nn-th order Farey sequence, let a/b,a/ba / b, a^{\prime} / b^{\prime} be the fractions adjacent to 1/21 / 2 on the left and right, respectively. Prove: b=b=1+2[(n1)/2]b=b^{\prime}=1+2[(n-1) / 2], i.e., bb is the largest odd number not exceeding nn, and a+a=ba+a^{\prime}=b.

Solution

9. Use inductive proof.

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