MathDB

5

Part of 2003 IMC

Problems(2)

IMC 2003 Problem 5

Source: IMC 2003

3/7/2021
Let g:[0,1]→Rg:[0,1]\rightarrow \mathbb{R} be a continuous function and let fn:[0,1]→Rf_{n}:[0,1]\rightarrow \mathbb{R} be a sequence of functions defined by f0(x)=g(x)f_{0}(x)=g(x) and fn+1(x)=1x∫0xfn(t)dt.f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt. Determine lim⁡n→∞fn(x)\lim_{n\to \infty}f_{n}(x) for every x∈(0,1]x\in (0,1].
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IMC 2003 Problem 11

Source: IMC 2003 Day 2 Problem 5

11/2/2020
a) Show that for each function f:Q×Q→Rf:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}, there exists a function g:Q→Rg:\mathbb{Q}\rightarrow \mathbb{R} with f(x,y)≤g(x)+g(y)f(x,y) \leq g(x)+g(y) for all x,y∈Qx,y\in \mathbb{Q}. b) Find a function f:R×R→Rf:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, for which there is no function g:Q→Rg:\mathbb{Q}\rightarrow \mathbb{R} such that f(x,y)≤g(x)+g(y)f(x,y) \leq g(x)+g(y) for all x,y∈Rx,y\in \mathbb{R}.
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