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Part of 2011 IMC

Problems(2)

IMC 2011 Day 1 Problem 1

Source:

7/30/2011
Let f:R→Rf:\mathbb{R} \to \mathbb{R} be a continuous function. A point xx is called a shadow point if there exists a point y∈Ry\in \mathbb{R} with y>xy>x such that f(y)>f(x).f(y)>f(x). Let a<ba<b be real numbers and suppose that ∙\bullet all the points of the open interval I=(a,b)I=(a,b) are shadow points; ∙\bullet aa and bb are not shadow points. Prove that a) f(x)≤f(b)f(x)\leq f(b) for all a<x<b;a<x<b; b) f(a)=f(b).f(a)=f(b).
Proposed by José Luis Díaz-Barrero, Barcelona
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IMC 2011 Day 2, Problem 1

Source:

8/1/2011
Let (an)⊂(12,1)(a_n)\subset (\frac{1}{2},1). Define the sequence x0=0,xn+1=an+1+xn1+an+1xnx_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}. Is this sequence convergent? If yes find the limit.
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