IMO Shortlist 2014 problem A2


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7. svibnja 2017.
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Define the function f : (0, 1) \to (0, 1) by f(x) = \begin{cases}
    x + \frac12 & \text{if } x < \frac12 \text{,} \\
    x^2 & \text{if } x \geq \frac12 \text{.}
  \end{cases} Let a and b be two real numbers such that 0 < a < b < 1. We define the sequences a_n and b_n by a_0 = a, b_0 = b, and a_n = f(a_{n-1}), b_n = f(b_{n-1}) for n > 0. Show that there exists a positive integer n such that (a_n - a_{n-1})(b_n - b_{n-1}) < 0 \text{.}

(Denmark)

Izvor: https://www.imo-official.org/problems/IMO2014SL.pdf