>>13823663One final remark is about the insight. I believe that insight is, in a way, part of resources. You might develop good control but sometimes, you come across a problem which needs you to observe a very specific fact or know a very specific concept. I remember that in IIT JEE, knowing L'Hospital's rule was necessary for a particular problem and the solution was like a one line solution which used this rule. Similarly, there are some specific theorems that are used in the IMO and not knowing them will basically mean that no matter how good a control you have, you will fail. The only use of control in this case is that you will know that you are missing something critical and will realize that you better give it up for the moment. Sometimes, it is not even a specific theorem. Certain heuristics can also become the critical elements in the solution to an IMO problem. The point of all this is that ultimately, lacking certain resources can cost you the problem. This point has been very nicely expanded in this book. He takes the example of a problem where you need to divide the triangle into two parts so that their areas are equal. This has to be accomplished by constructing a line parallel to one of the sides of the triangle. This is much simpler than an IMO problem and is perhaps just an exercise in geometry; however, it is a good problem to make the point. In this problem, it is tough to obtain the solution unless you use the fact that the line constructed will result in the formation of another triangle similar to the main triangle. Even though it is obvious, this point has to be USED in the solution or else, it will be very hard to figure out the solution.
On a much tougher scale, you will see the EXACT same thing happening in the IMO. You HAVE to know certain theorem, heuristic etc to cract certain problems. Hence the advice to practice a lot makes sense.
