2016 2016 2016
2 0 1 6 y 2 0 1 6 x + 2 0 1 5 times 2 0 1 6 2 + 2 0 1 6 2 + … + 2 0 1 6 2 = 2 0 1 6
where x and y are both positive integers
What is the value of 2 0 1 6 x y x 3 − y 3 + 2 0 1 6 x y ?
The answer is 1.
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1 solution
Great solution +1. I hope there's no another possible solution.
2 0 1 6 y 2 0 1 6 x + 2 0 1 5 times 2 0 1 6 2 + 2 0 1 6 2 + … + 2 0 1 6 2 = 2 0 1 6 2 0 1 6 x + 2 0 1 5 ( 2 0 1 6 2 ) = 2 0 1 6 ( 2 0 1 6 y ) 2 0 1 5 ( 2 0 1 6 2 ) = 2 0 1 6 1 + y − 2 0 1 6 x 2 0 1 5 = 2 0 1 6 2 2 0 1 6 1 + y − 2 0 1 6 x 2 0 1 6 − 1 = 2 0 1 6 y − 1 − 2 0 1 6 x − 2 2 0 1 6 1 − 2 0 1 6 0 = 2 0 1 6 y − 1 − 2 0 1 6 x − 2
Compare the terms, and we get
y − 1 = 1 ⟹ y = 2 x − 2 = 0 ⟹ x = 2
Therefore,
2 0 1 6 x y x 3 − y 3 + 2 0 1 6 x y = 2 0 1 6 ( 2 ) ( 2 ) 2 3 − 2 3 + 2 0 1 6 ( 2 ) ( 2 ) = 2 0 1 6 ( 4 ) 2 0 1 6 ( 4 ) = 1