What is a search problem?
A problem is a search problem if there's an algorithmic way to verify the answer.  In particular, let $V$ be any algorithm that takes two inputs and always terminates.  Then the following is a search problem:

Input: $x$
Goal: find and output any $y$ such that $V(x,y)=1$, or report that no such $y$ exists

Every search problem can be written in this form.  In particular, you can take the above as the definition of a search problem: a problem is a search problem if there exists an always-terminating algorithm $V$ that has the above form.