Advice: How can one prepare for a maths entrance exam? How can one develop mathematical thinking? I am Computer science student so pardon me if I am asking this in a wrong place!
My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not justifying my computer science degree, I am  trying to work on my rusty math since last two years but until now I am not much confident enough the main problem that I am facing is that I no matter whatever I do I am making silly mistakes, I will be presenting a college level maths entrance exams next year, and am preparing for this for two years; but now there is a feeling that I might lack ingenuity, which is required to be able to crack any competitive exam. So, is there any way to develop mathematical thinking? Does a mathematician usually/(at some part of their life) follow any exercise to concentrate?
I would also mention that I am almost alone in my home and I am rather compelled to learn maths all my own. But is this even  possible? I have taken some correspondence classes and I am trying with all my efforts but is it really possible to learn on my own? Please help me!
 A: Whether or not "ingenuity" is "required to be able to crack" (do well?) the exam you will present depends enormously on the kind of exam it is. Ingenuity is not particularly required, say, in the United States' GRE examination (at least, when I took it, even the Mathematics Subject test). 
The ability to concentrate has little to do with mathematics per se; it is a requirement of any serious field of study or endeavor. 
The best way to develop "ingenuity" (really, ability to solve certain kinds of problems) is practice. Practice, practice, practice. And then, more practice. The vast majority of problems you will encounter are problems that have something in common with other problems, that can be analyzed with the same kind of ideas, the same kind of methods. So what you really want is not ways to concentrate, or ways to become "ingenious", but a big "portfolio" of problems that you know how to solve and that you can refer to when you encounter a new problem. 
There is few advice better than that given by Polya decades ago in his book How to Solve It. See some of the reviews there for some of the questions to ask. And always ask: "Is this like something I've seen before? If so, can I try doing things similar to what I did then?"
Added: Yes, it is possible to learn on your own, though it is always much easier to learn with immediate feedback and help from others. From your description of the exam ("only 30 people selected in all of India") things may be very hard even if you had immediate feedback and the best tutor money could rent available 24 hours a day at your beck and call. If what you want is reassurance that you are doing things correctly when you are in doubt, there are sites such as this one that can help; try not to get carried away and flood the site with all the questions that have accummulated over the past two years (you'll turn off people), but reasoned questions showing both interest and that you've tried to work them through on your own and are genuinely confused are likely to be well received. None of this, however, has to do with developing "ingenuity", or the ability to concentrate, or "mathematical thinking". 
A: There are plenty of good resources available on the 'net in form of lecture notes, often with homework and exams and the corresponding solutions. Try your hand at the problems, compare with the proposed solutions. Ask here in case of confusion/discrepancy.
The preparation material for the math olympiads, their problems and solutions, may help in getting the hang of tackling complex problems, looking for short solutions.
Pólya's "How to solve it" is required reading. Check out Hammack's "Book of proof", and texts on math writing, like Knuth et al's "Mathematical writing". Those are skills that are taken for granted (how anybody in their right mind can think this is inborn is beyond me). Practice your LaTeX, you will breathe it all day long.
Rummage around, the Internet contains many valuable resources. But don't forget Sturgeon's law...
