MathDB

24

Part of 2019 India PRMO

Problems(2)

Combinatorics / Counting squares

Source:

8/11/2019
A 1×n1 \times n rectangle (n≥1n \geq 1 ) is divided into nn unit (1×11 \times 1) squares. Each square of this rectangle is colored red, blue or green. Let f(n)f(n) be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of f(9)/f(3)f(9)/f(3)? (The number of red squares can be zero.)
combinatoricsgeometryrectangle
9's in a Number

Source: PRMO Leg 2 P24

8/25/2019
For n≥1n \geq 1, let ana_n be the number beginning with nn 99's followed by 744744; eg., a4=9999744a_4=9999744. Define f(n)=max{m∈N∣2m divides an}f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}, for n≥1n\geq 1. Find f(1)+f(2)+f(3)+⋯+f(10)f(1)+f(2)+f(3)+ \cdots + f(10).
number theoryPRMO