UKMT - Senior Mathematical Challenge - 2022 - 24 | Exact Science

24

The numbers x, y, z, and w are all integers. x and y are variable, and z and w are constant and positive. The four integers are related by the equation xy = z(x + w). When y takes its maximum possible value, which expression is equal to y - x?

A. zw - 1 B. (z - 1)(w - 1) C. (z + 1)(w - 1) D. (z - 1)(w + 1) E. (z + 1)(w + 1)

Answer

D

SOLUTIOn

Maximizing H requires minimizing G−@, leading to G−@ = 1. Substituting into the equation gives H − G = (?−1)(@+1).

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