Degree 99 Polynomial

Algebra Level 5

f(x)f(x) is a polynomial of degree 99. For exactly 100 (out of 101) integer values ranging from 0 0 to 100 100, we have f(x)=1x+1 f(x) = \frac {1}{x+1} . Also, f(101)=0f(101) = 0. For what value of aa, 0a100 0 \leq a \leq 100 is f(a)1a+1 f(a) \neq \frac {1}{a+1} ?

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