MathDB

4

Part of 1998 IMC

Problems(2)

prove that there exists \xi

Source: IMC 1998 day 1 problem 4

11/1/2005
The function f:R→Rf: \mathbb{R}\rightarrow\mathbb{R} is twice differentiable and satisfies f(0)=2,f′(0)=−2,f(1)=1f(0)=2,f'(0)=-2,f(1)=1. Prove that there is a ξ∈]0,1[\xi \in ]0,1[ for which we have f(ξ)⋅f′(ξ)+f′′(ξ)=0f(\xi)\cdot f'(\xi)+f''(\xi)=0.
functioncalculusderivativereal analysisreal analysis unsolved
IMC 1998 Problem 10

Source: IMC 1998 Day 2 Problem 4

10/28/2020
Let Sn={1,2,...,n}S_{n}=\{1,2,...,n\}. How many functions f:Sn→Snf:S_{n} \rightarrow S_{n} satisfy f(k)≤f(k+1)f(k) \leq f(k+1) and f(k)=f(f(k+1))f(k)=f(f(k+1)) for k<n?k <n?
functioncombinatorics