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Integer points z = a + ib inside a region of the complex plane

This type of question, asked 2 times: 28 Jun 2022, 9 Apr 2024. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    28 Jun 2022

    89
    The number of elements in the set {z=a+ib∈C:a,b∈Z\{ z = a + ib \in \mathbb{C}:a,b \in \mathbb{Z} and 1<∣z−3+2i∣<4}1 < |z - 3 + 2i| < 4\} is

  2. Question 2

    9 Apr 2024

    82
    The sum of the square of the modulus of the elements in the set
    {z=a+ib:a,b∈Z,z∈C,∣z−1∣≤1,∣z−5∣≤∣z−5i∣}\mathbf{\{ z = a + ib:a,b \in Z,z \in C,|z - 1| \leq 1,|z - 5| \leq |z - 5i|\}}
    is