Current set of maths online problems for BStat (ISI entrance exam, India) and other mathematical contests. (solutions later). Sample problems are given here, exclusive 100+ other problems available after paying 1$. Drop me a comment if you want the questions (and answers).
Note: I have solutions to a lot of these questions, almost all. But, it could be that some solution slipped by time, since I used these long time ago. In that case, I might not have some solutions.
*1. Consider a `4n` * `4n` square. Now, consider rectangles of integer dimensions which would fit inside this square (max dimension of them can be `4n`). What is the probability (considering very large `n`), that these rectangles have an area lesser than or equal to `4n^{2}` chosen over all possible dimensions (that would fit inside the square).
*2. Prove that, if a number `n` has the prime number expansion form of `p_{1}^{k_{1}}*p_{2}^{k_{2}}*p_{3}^{k_{3}}...`, the sum of its factors is (where `1` and `n` are also considered factors) is given by `(p_{1}^{k_{1}+1} - 1) / (p_{1} - 1) * (p_{2}^{k_{2}+1} - 1) /(p_{2} - 1) * ...`
*3. From the above definition of factors, prove that a perfect number has sum of its factors, equal to twice the number.
**4. Prove fermat's little theorem: If `p` is a prime, `a^{p} \equiv a (mod p)`. (Hint: Use binomial expansion to power p, and mathematical induction on a).
5. Prove that if `p` is not a prime, then `2^{p} - 1` is not a prime.
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