# 1000 Degree Polynomial

Algebra Level 5

How many polynomials $f(x)$ are there, which satisfy all the following conditions:

1. $f(x)$ is monic.
2. $f(x)$ has degree 1000.
3. $f(x)$ has real coefficients.
4. $f(x)$ divides $(x-1)f(2x)$.
5. $f(2)$ is an integer.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial $x^3 + 3x - 5$ is monic but the polynomial $-x^4 + 2x^3 - 6$ is not.

As stated in the problem, the coefficients are real values, not necessarily integer valued.

Clarification: The divisibility condition in 4 is about polynomials, not integer values obtained upon evaluation. For example, $x-1$ divides $x^2 -1$. since $x^2-1 = (x-1)(x+1)$. $x-2$ does not divide $x^2-1$, even though when "$x=3$", 1 divides 8.

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