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Algebra Level 4

f ( 299 + x ) = f ( 299 − x ) \Large{f(299+x) = f(299-x)}

Let f ( x ) f(x) be a function which satisfies the above functional equation for all real values of x x . If f ( x ) f(x) has exactly three roots, say α , β , γ \alpha, \beta, \gamma , find the value of α + β + γ \alpha + \beta +\gamma .


The answer is 897.

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Satyajit Mohanty by Satyajit Mohanty , 24, India

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1 solution

Bufang Liang
Aug 27, 2015

Notice that the function is symmetric around x=299. If one of the roots is less than 299 (specifically 299-x), then there exists a corresponding root greater than 299 (specifically 299+x). Therefore, 299 itself must be a root or else you cannot have an odd number of roots.

299+299-x+299+x = 299*3 = 897

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But there may be a repeated root at 299

Atul Solanki - 5 years, 8 months ago

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