Proof Writing & Problem Solving · Superchallenge problem

Claim: if a polynomial takes an integer value at every integer, then all its coefficients are integers. This is false. How many polynomials $p(x) = ax^3 + bx^2 + cx + d$ with EACH of $a, b, c, d$ in the interval $0 \le t < 1$ take an integer value at every integer $x$?