MathDB

3

Part of 2019 Romania Team Selection Test

Problems(2)

Number of fixed points of certain condition functions

Source: American Mathematical Monthly

10/1/2019
Let be three positive integers a,b,c a,b,c and a function f:N⟶N f:\mathbb{N}\longrightarrow\mathbb{N} defined as f(n)={n−a,n>cf(f(n+b)),n≤c. f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . Determine the number of fixed points this function has.
functionalgebra
Special Rectangular Grids

Source: Romania TST 2019 Day 5 P3, based on Rioplatense 2010 Level 2

3/5/2022
Given an integer n≥2,n\geq 2, colour red exactly nn cells of an infinite sheet of grid paper. A rectangular grid array is called special if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly nn red cells, let NN be the largest number of red cells a special rectangular grid array may contain. Determine the least value NN may take over all possible configurations of exactly nn red cells
combinatoricsgridcolouring