The era of searching on the internet has increased the ease of looking up answers and solutions to textbook questions, and so perhaps it has increased the percentage of students who go to the trouble of looking for a shortcut rather than working out the problem on their own. But I do not think that it is a new problem.
There have always been resources to turn to which had solutions to problems:
copies of the teacher's editions (for the elementary algebra, calculus, and physics classes)
workbooks which provide problems and detailed solutions
archives maintained by groups (fraternities, sororities, societies such as student mathematics groups or physics clubs or biomedical engineering clubs)
solution manuals are often published to go along with a text, either by the same publisher or a competitor
geometry has always (in the last few centuries) been taught in a programmatic step by step fashion, with proofs building up on proofs. There are multiple centuries (millennia) of history and texts to look at and study from. Sometimes simply by looking ahead in the book, it's possible to get a clear answer on what the structure and details of a proof ought to include.
In all of these cases, the students are either (1) cheating themselves out of exercising their brains and coming up with a solution on their own, or (2) helping themselves past a hurdle which they could not overcome on their own and which they've decided to bypass by taking someone else's answer. Only the student can know if they've spent hours or days working at it and found it too frustrating, or not worth the effort of waiting on it / sleeping on it / approaching it again on another day.
So, yes, the internet and search engines have increased the fraction of students who might use a shortcut instead of doing the work themselves. However, the good students (the ones who would want to go on to solve problems on their own initiative, or try to solve the same ones again in different ways, perhaps even become mathematicians and scientists) will probably not be the ones who would takes those shortcuts and bypasses around the obstacles. Hmmm, I realize this is sounding like I'm saying not to care about the ones who would cheat. The problem with teaching, and caring about teaching, is that a teacher would like students who want to learn and find joy in knowledge and problem solving. However, teachers do not get to choose their students, as they did back in the days of Hippocrates when doctors literally could choose or refuse to teach particular students. Teachers get the students who sign up for their classes. Teachers cannot improve the motivation or attitude of their students. Students have to be responsible for their own education at some point. We can provide lessons or be the water; the horse has to actually drink.