An algebra problem by A Former Brilliant Member
If S is the sum of all roots of the equation x 2 0 1 7 + ( 2 1 − x ) 2 0 1 7 = 0 , then find the sum of the digits of S .
The answer is 9.
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2 solutions
l will "recycle" an old solution :
Make a substitution x = y + 4 1 and write the equation as ( 4 1 + y ) 2 0 1 7 + ( 4 1 − y ) 2 0 1 7 = 0 . The sum of the roots of this even polynomial of degree 2016 is 0, so that the sum of the roots of the original polynomial is S = 4 2 0 1 6 = 5 0 4 . The sum of the digits of S is 9
x 2 0 1 7 + ( − x 2 0 1 7 − ( 1 2 0 1 7 ) × 0 . 5 × x 2 0 1 6 + ( 2 2 0 1 7 ) × ( 0 . 5 ) 2 × x 2 0 1 5 . . . . . . . Hence sum of roots= ( 1 2 0 1 7 ) × ( 0 . 5 ) 2 ( 2 2 0 1 7 ) × 0 . 5 =504. Sum of digits = 9