Let a > 2 be a constant. If there are just 18 positive integers satisfying the inequality (x−a)(x−2a)x−a2<0, then the value of a is ________.

As a > 2,a2>2a>a>2Now, (x−a)(x−2a)x−a2<0Thus, the solution set is as shown. Between 0, a, there are (a - 1) positive integers and between 2a, a2, there are a2−2a−1 integers. Therefore, a2−2a−1+a−1=18or a2−a−20=0or (a−5)(a+4)=0∴ a=5