(Canadian Senior Mathematics Contest 2022, Part B, Question 2, CEMC - UWaterloo)
Adapted from original statement.

a) Determine all real numbers $a \gt 0$ for which $\sqrt{a^2+a} = \frac{2}{3}$.
b) For each positive integer $m$, determine the difference between $(m + \frac{1}{2})^2 + (m + \frac{1}{2})$ and the nearest perfect square.
c) For every positive integer $n$, prove that the number of positive integers $c$ with $n \lt \sqrt{c + \sqrt{c}} \lt n + 1$ is even. (do not answer)

Answer Submission Note(s)
Please do not include the answer for the final part when submitting. It is only included here for integrality.
Fractions should be typed as "x/y" and the final answer should have answers for the first two parts in order and space-separated.