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Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

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Largest Expression

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Given that $x$ is positive and less than $1$, we may deduce that
$$x^2+x > x^2+x^3 > x^2 > x^3 > x^4$$

This problem is taken from the UKMT Mathematical Challenges.