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

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

The answer is 897.

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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