Starting from this year, the mathematics Olympiad candidates in India will have to give a Pre Regional Mathematical Olympiad (PRMO) before the actual Regional Mathematical Olympiad (RMO) later. The 2017 PRMO conducted by the Indian Association of Physics Teachers (IAPT) this year. The questions are...

 Find all integers $$ a,b,c $$ satisfying $$1<a<b<c$$ such that $$(a-1)(b-1)(c-1)$$ is a divisor of $$abc-1$$. Find the number of ordered pairs $$(x,y)$$ of positive integers which satisfy $$xy=27027$$. The integer $$N$$ consists of 2017 consecutive nines. A computer calculates $$N^3 = (999 ....

For which $$n\in \mathbb{N}$$, does $$n+1 | \binom{2n}{n}$$ hold? For a polynomial $$p(x) = a_nx^n+\ldots + a_0$$ with integral coefficients, i.e. $$a_i\in \mathbb{Z}$$ for all $$1\leq i\leq n$$ with $$a_n\neq 0$$, if $$p(\frac{r}{s})=0$$ where $$r$$, $$s$$ are coprime integers with $$s\neq 0$$ then show that: $$r|a_0$$ ...

Given a straight line $$l$$ and points $$A$$ and $$B$$ on the same side of it find the shortest path from $$A$$ to $$B$$ which touches $$l$$. Given a function $$f:\mathbb{Q}\rightarrow\mathbb{Q}$$ such that $$f(x+y)=f(x)+f(y)$$ Show that $$f(x)=xf(1)$$ for all $$x\in\mathbb{Q}$$. Thus, infer that for such a...