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HKIMO 2019 THPT (Grades 10–12 shared) National Final (Heat) Set 02 Question 7

HKIMO · HKIMO 2019 THPT (Grades 10–12 shared) National Final (Heat) Set 02

Question 7

If $x=5-\dfrac{18}{2+\dfrac1{5-\dfrac{18}{2+\cdots}}}$, find the minimum value of x.

Editorial interpretation: the repeated fraction is treated as its self-similar fixed-point equation; no particular sequence of finite truncations is specified.