MathDB

4

Part of 2009 IberoAmerican Olympiad For University Students

Problems(1)

Slippery functions - OIMU 2009 Problem 4

Source:

5/23/2010
Given two positive integers m,nm,n, we say that a function f:[0,m]→Rf : [0,m] \to \mathbb{R} is (m,n)(m,n)-slippery if it has the following properties:
i) ff is continuous; ii) f(0)=0f(0) = 0, f(m)=nf(m) = n; iii) If t1,t2∈[0,m]t_1, t_2\in [0,m] with t1<t2t_1 < t_2 are such that t2−t1∈Zt_2-t_1\in \mathbb{Z} and f(t2)−f(t1)∈Zf(t_2)-f(t_1)\in\mathbb{Z}, then t2−t1∈{0,m}t_2-t_1 \in \{0,m\}.
Find all the possible values for m,nm, n such that there is a function ff that is (m,n)(m,n)-slippery.
functionreal analysisreal analysis unsolved