Function in Function in Function in Function in Function

Algebra Level pending

Let f : Z + × Z → Z f: \mathbb{Z}^{+} \times \mathbb{Z} \rightarrow \mathbb{Z} such that

  1. f ( 0 , y ) = − y f(0,y) = -y
  2. f ( x + 1 , y ) = f ( x , f ( x , f ( x , f ( x , f ( x , − y ) ) ) ) ) f(x+1,y) = f(x,f(x,f(x,f(x,f(x,-y)))))

Find the value of f ( 31415 , 92653 ) f(31415,92653)

This problem is adapted from Thailand 8th Mathematics POSN (1st round preceding 2nd round).


The answer is 92653.

Samuraiwarm Tsunayoshi by Samuraiwarm Tsunayoshi , 22, Thailand

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1 solution

Here's the first solution. I might come up with the second one.

Substituting x → 0 , 1 x \rightarrow 0, 1 we get

f ( 1 , y ) = f ( 0 , f ( 0 , f ( 0 , f ( 0 , f ( 0 , − y ) ) ) ) ) = y f(1,y) = f(0,f(0,f(0,f(0,f(0,-y))))) = y

f ( 2 , y ) = f ( 1 , f ( 1 , f ( 1 , f ( 1 , f ( 1 , − y ) ) ) ) ) = − y f(2,y) = f(1,f(1,f(1,f(1,f(1,-y))))) = -y .

Now we're proving that f ( n ) = f ( n + 2 k ) f(n) = f(n+2k) for any integers k , n k,n such that n ≥ 0 n \geq 0 .

Let f ( n , y ) = ± y f(n,y) = \pm y

f ( n + 1 , y ) = f ( n , f ( n , f ( n , f ( n , f ( n , − y ) ) ) ) ) = ∓ y f(n+1,y) = f(n,f(n,f(n,f(n,f(n,-y))))) = \mp y

f ( n + 2 , y ) = f ( n + 1 , f ( n + 1 , f ( n + 1 , f ( n + 1 , f ( n + 1 , − y ) ) ) ) ) = ± y f(n+2,y) = f(n+1,f(n+1,f(n+1,f(n+1,f(n+1,-y))))) = \pm y

Proving by induction, if f ( n + 2 a , y f(n+2a,y = y), then f ( n + 2 a + 2 , y ) = ± y f(n+2a+2,y) = \pm y .

f ( n + 2 a + 1 , y ) = ∓ y f(n+2a+1,y) = \mp y

f ( n + 2 a + 2 , y ) = ± y f(n+2a+2,y) = \pm y

Therefore, f ( n ) = f ( n + 2 k ) f(n) = f(n+2k) .

f ( 31415 , 92653 ) = f ( 1 + 2 × 15707 , 92653 ) = 92653 f(31415,92653) = f(1+2\times 15707,92653) = \boxed{92653} . ~~~

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2nd solution

I'll let you prove that f ( x , f ( x , f ( x , y ) ) ) = f ( x , y ) f(x,f(x,f(x,y))) = f(x,y) lol

Therefore, f ( x + 1 , y ) = f ( x , − y ) f(x+1,y) = f(x,-y) .

f ( 31415 , 92653 ) = 92653 f(31415, 92653) = \boxed{92653} heheheee

Samuraiwarm Tsunayoshi - 6 years, 11 months ago

I'm sorry for the wrong wording. f f can only accept positive integers and non-negative integers.

Samuraiwarm Tsunayoshi - 6 years, 11 months ago

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If f f can only accept positive integers in the x-coordinate, why do we have f ( 0 , y ) = − y f(0,y) = -y ?

I think you want f : Z ≥ 0 × Z → Z f: \mathbb{Z} ^{ \geq 0 } \times \mathbb{Z} \rightarrow \mathbb{Z} initially , and then to state in condition 2 that x ≥ 0 , y ∈ Z x \geq 0, y \in \mathbb{Z} .

Calvin Lin Staff - 6 years, 11 months ago

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