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Romanian District Olympiad 2019 - Grade 11 - Problem 4

Source: Romanian District Olympiad 2019 - Grade 11 - Problem 4

March 16, 2019
continuous functionConvergenceSequencescalculus

Problem Statement

Let f:[0,∞)→[0,∞)f: [0, \infty) \to [0, \infty) be a continuous function with f(0)>0f(0)>0 and having the property x−y<f(y)−f(x)≤0 ∀ 0≤x<y.x-y<f(y)-f(x) \le 0~\forall~0 \le x<y. Prove that: a)a) There exists a unique α∈(0,∞)\alpha \in (0, \infty) such that (f∘f)(α)=α.(f \circ f)(\alpha)=\alpha. b)b) The sequence (xn)n≥1,(x_n)_{n \ge 1}, defined by x1≥0x_1 \ge 0 and xn+1=f(xn) ∀ n∈Nx_{n+1}=f(x_n)~\forall~n \in \mathbb{N} is convergent.
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