These three problems can all be solved using a similar sort of method: but it's probable that other methods exist! We're interested to see your ideas.

1. Find all pairs of integer solutions (x,y) to the equation x²-2x-y²=2.
2. Find all pairs of integer solutions (a,b) to the equation 4a²+4a = b²+b.
3. Find all pairs of integer solutions (p,q) to the equation p²+3 = q(p+2).