If the teachers are so good at these exams that they can flash-solve them, why aren't they doctors? Surely that pays better than teaching at a cram school?
Teachers typically specialize in a single subject and tests are multi-disciplinary. The subjects can range from literature to chemistry to history to physics. As in, you have to be at the top of the pack for literature questions even if you're applying for engineering. And recall, these are no pushover questions; I've taken some of these elite school exams as well as the american SAT/SAT 2 and the SATs are child's play in comparison.
Also, competitiveness for some disciplines is seriously no joke: to give you an idea, medicine at Harvard has an admission rate of around 3.5%. Medicine at University of Sao Paulo, Brazil has an admission rate of 0.8%. Admission rates for IIT, India are even worse, at around 0.59%.
And if that isn't discouraging enough, many of these schools have reputations of being "hard to get in, harder to get out" (meaning that actually graduating is ever harder than getting accepted into the program in the first place)
