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CSMC 2018 Part B Problem 2

Source:

November 24, 2018
CSMCCSMC 2018

Problem Statement

[*]Determine the positive integer xx for which 14−1x=16.\dfrac14-\dfrac{1}{x}=\dfrac16. [*]Determine all pairs of positive integers (a,b)(a,b) for which ab−b+a−1=4.ab-b+a-1=4. [*]Determine the number of pairs of positive integers (y,z)(y,z) for which 1y−1z=112.\dfrac{1}{y}-\dfrac{1}{z}=\dfrac{1}{12}. [*]Prove that, for every prime number pp, there are at least two pairs (r,s)(r,s) of positive integers for which 1r−1s=1p2.\dfrac{1}{r}-\dfrac{1}{s}=\dfrac{1}{p^2}.
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