A problem from RMO 2006

Algebra Level 3

Let X X denote the set of all natural numbers greater than or equal to 8. Let f : X → X f :X\rightarrow X be a function that satisfies f ( x + y ) = f ( x y ) f(x+y)=f(xy) for all positive integers x x and y y greater than or equal to 4. If f ( 8 ) = 9 f(8)=9 , then find the value of f ( 9 ) f(9) .


The answer is 9.

Rakhi Bhattacharyya by Rakhi Bhattacharyya , 49, India

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1 solution

What it needs is observation in solving problems like this . For all x ≥ 4 , y ≥ 4 x\ge4,y\ge4 we have ,

Using f ( x + y ) = f ( x y ) f(x+y)=f(xy)

f ( 8 ) = f ( 4 + 4 ) = f ( 4.4 ) = f ( 8 + 8 ) = f ( 8.8 ) = f ( 16.4 ) = f ( 16 + 4 ) = f ( 20 ) = f ( 4.5 ) = f ( 4 + 5 ) = f ( 9 ) \large f(8) = f(4+4)=f(4.4) = f(8+8)=f(8.8)=f(16.4)=f(16+4)=f(20)=f(4.5)=f(4+5)=f(9)

So f ( 9 ) = f ( 8 ) = 9 f(9)=f(8)=9 using the relations and it's valid as all x , y ≥ 4 x,y\ge4 are used.

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