MathDB
Recursively Defined Functions

Source: 2012 AIME II Problem 11

March 29, 2012
functionAMCAIMEnumber theoryrelatively primeAIME IInice words

Problem Statement

Let f1(x)=23−33x+1f_1(x) = \frac{2}{3}-\frac{3}{3x+1}, and for n≥2n \ge 2, define fn(x)=f1(fn−1(x))f_n(x) = f_1(f_{n-1} (x)). The value of x that satisfies f1001(x)=x−3f_{1001}(x) = x - 3 can be expressed in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.
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