1000 Degree Polynomial

Algebra Level 5

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

  1. f(x)f(x) is monic.
  2. f(x)f(x) has degree 1000.
  3. f(x)f(x) has real coefficients.
  4. f(x) f(x) divides (x1)f(2x) (x-1)f(2x) .
  5. f(2) f(2) is an integer.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -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, x1x-1 divides x21x^2 -1. since x21=(x1)(x+1)x^2-1 = (x-1)(x+1). x2x-2 does not divide x21x^2-1, even though when "x=3x=3", 1 divides 8.

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