Try this beautiful problem from the Pre-RMO, 2019 based on Rectangle and Squares.
Rectangles and squares – PRMO 2019
A \(1 \times n\) rectangle \(n \geq 1\) is divided into n unit \( 1 \times 1 \) squares. Each square of this rectangle is coloured red, blue or green. Let f(n) be the number of colourings of the rectangle in which there are an even number of red squares, find the largest prime factor of \(\frac{f(9)}{f(3)}\)
- is 107
- is 37
- is 840
- cannot be determined from the given information
Key Concepts
Combinations
Algebra
Integers
Check the Answer
But try the problem first…
Answer: is 37.
PRMO, 2019, Question 24
Combinatorics by Brualdi
Try with Hints
First hint
\(f(n)\)=\({n \choose 0}2^{n} + {n\choose2} 2^{n-2} + {n\choose 4} 2^{n-4}+…..\) and \((2+1)^{n}\)=\({n\choose0} 2^{n}+ {n\choose1} 2^{n-1} + {n\choose2} 2^{n-2} +…..\) and \((2-1)^{n}\)= \({n\choose0} 2^{n}-{n\choose1} 2^{n-1}+{n\choose2} 2^{n-2}-…..\)
Second Hint
adding gives \(\frac{3^{n}+1}{2}=f(n)\) \(\frac{f(9)}{f(3)}=\frac{3^{9}+1}{3^{3}+1}=3^{6}-27+1\)=703
Final Step
then 703=\(19 \times 37\) then largest factor =37.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA
Related Program
Math Olympiad Program… taught by Olympians and researchers