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April 13, 2020

Rectangle and Squares | PRMO 2019 | Question 24

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


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.

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