Nowhere island

Algebra Level 2

The mathematicians in Nowhere island introduced a new operation. An islander told us that the operation's symol is ♣ \clubsuit . Furthermore he told us that for every x , y , z x, y, z numbers

  • x ♣ x = 0 x\clubsuit x=0

  • x ♣ ( y ♣ z ) = ( x ♣ y ) + z x\clubsuit(y\clubsuit z)=(x\clubsuit y)+z

Find the value of 2017 ♣ 7 2017\clubsuit7 .

Note : The ♣ \clubsuit is assign x ♣ y x\clubsuit y for every ( x ; y ) (x;y) number pair.


The answer is 2010.

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

x ♣ ( y ♣ y ) = x ♣ 0 = x ♣ ( x ♣ x ) = ( x ♣ x ) + x = x x\clubsuit(y\clubsuit y)=x\clubsuit0=x\clubsuit(x\clubsuit x)=(x\clubsuit x)+x=x

x ♣ ( y ♣ y ) = ( x ♣ y ) + y x\clubsuit(y\clubsuit y)=(x\clubsuit y)+y

⇒ ( x ♣ y ) + y = x \Rightarrow (x\clubsuit y)+y=x

⇒ x ♣ y = x − y \Rightarrow x\clubsuit y=x-y

⇒ 2017 ♣ 7 = 2010 \Rightarrow 2017\clubsuit7=\boxed{2010}

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