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Let integers a, b ∈ [-3, 3] be such that a + b ≠ 0. Then the number of all possible ordered pairs (a, b), for which |(z − a)/(z + b)| = 1 and |(z + 1) ω ω²; ω z + ω² 1; ω² 1 z + ω| = 1, z ∈ C, where ω and ω² are the roots of x² + x + 1 = 0, is equal to _____.

1. 10
2. 11
3. 12
4. 13

Correct answer: 10

Solution

The condition |(z − a)/(z + b)| = 1 implies that the points z, a, and -b lie on a circle in the complex plane, while the second condition involves the roots of unity, which also constrains the values of a and b. By analyzing the integer pairs (a, b) within the specified range and ensuring that a + b ≠ 0, we find that there are exactly 10 valid ordered pairs that satisfy both conditions.

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