Let $$n$$ be an odd integer. If $$\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $$ for every value of $$\theta ,$$ then

If the vertices $$P, Q, R$$ of a triangle $$PQR$$ are rational points, which of the following points of the triangle $$PQR$$ is (are) always rational point(s)?

If $$\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)$$ and $$S\left( {a,b} \right)$$ are the vertices of a parrallelogram $$PQRS,$$ then

Let $$p$$ be a prime and $$m$$ a positive integer. By mathematical induction on $$m$$, or otherwise, prove that whenever $$r$$ is an integer such that $$p$$ does not divide $$r$$, $$p$$ divides $${}^{np}{C_r},$$

[Hint: You may use the fact that $${\left( {1 + x} \right)^{\left( {m + 1} \right)p}} = {\left( {1 + x} \right)^p}{\left( {1 + x} \right)^{mp}}$$]