If $z\bar{z} = 1$, then the value of $\left|2 + \frac{1}{z}\right| + |2 - z|^2$ is ____.
Step-by-Step Solution
Key Concept: Transform the inequality into a homogeneous equation by substitution, then count non-negative integer solutions case by case using combinations.
Given $a+b+3c ≤ 30$ with $a, b, c$ positive integers, substitute $a=a_1+1$, $b=b_1+1$, $c=c_1+1$ to get $(a_1+1)+(b_1+1)+3(c_1+1)+d=30$, simplifying to $a_1+b_1+3c_1+d=25$. Since the coefficient of $c_1$ is 3, it cannot exceed 8, yielding 9 cases. For each value of $c_1$ from 0 to 8, find non-negative integer solutions using the stars and bars formula $C_{n+k-1}^{k-1}$. Summing all cases: $351+276+210+153+105+66+36+15+3 = 1215/5 = 243$.
Correct Answer: 10