I have been at TIFR for a week. Some thoughts:
The emphasis is clearly on “what book have you read? I followed this book. Have you attempted the exercises from this book?” This is similar to what I saw at ISI, Kolkata. I did not focus on problem solving while learning Mathematics on my own. My emphasis was on extrapolating definitions into unknown territory. I wanted to get a feel for the definitions. This proved to be a handicap in some ways as I could never quite get a hang of the techniques and machinery which would later help me understand deeper problems. Solving problems would have given me a sense of direction: it would give me a feel for which direction the mathematical community has taken after formulating these definitions (actually in most cases, before formulating those definitions).
However, I feel not solving problems has also been a major advantage. Now I know there is a world outside of problem solving, and hence aligning your outlook and interests with the rest of the mathematical community. I have the benefit of an outsider’s view. I have tried to go where no one else thought of going. If everything works out well, I might also have an insider’s view soon (after having worked through tons of problems in classic textbooks, of course). The future doesn’t look as bleak as I initially thought it would.