If x , y ∈ Z + and
y 3 = x 3 + 8 x 2 − 6 x + 8
Find the value of x + y .
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Nice Problem !
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Thanks! BTW, I too did the same! It is an RMO problem
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How can we say that no more solutions are possible? Thanks.
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@Satvik Golechha – I included an argument that explained this in my solution.
I did the same
Note that y 3 − ( x + 1 ) 3 = 5 x 2 − 9 x + 7 is always positive (check the discriminant). And y 3 − ( x + 3 ) 3 = − x 2 − 3 3 x − 1 9 is always negative. So the only possibility is that y = x + 2 . Plugging in and solving gives 2 x 2 − 1 8 x = 0 ; the only positive solution is x = 9 , so y = 1 1 and the answer is 2 0 .
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Here is what I did
The equation can be written as
y 3 − 2 3 = x 3 + 8 x 2 − 6 x
( y − 2 ) ( y 2 + 4 + 2 y ) = x ( x 2 + 8 x − 6 )
Now Let y − 2 = x . . . . . ( 1 ) y 2 + 4 + 2 y = x 2 + 8 x − 6 . . . . . ( 2 )
Substitute y = x + 2 in eq.(2) to get
x 2 + 6 x + 1 2 = x 2 + 8 x − 6 2 x = 1 8
By this we get x = 9 and y = 9 + 2 = 1 1
So x + y = 2 0