Question 11
Prove or disprove: For every integer $n \gt 0$, there is a polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ satisfying the following conditions:
(1) All coefficients $a_n, a_{n-1}, \dots, a_1, a_0$ are positive real numbers;
(2) At least one of the coefficients is $\frac{1}{n}$;
(3) The value of the polynomial is integer for every integer $x$.
(1) All coefficients $a_n, a_{n-1}, \dots, a_1, a_0$ are positive real numbers;
(2) At least one of the coefficients is $\frac{1}{n}$;
(3) The value of the polynomial is integer for every integer $x$.