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...

The 57th International Mathematical Olympiad was recently held in Hong Kong. The IMO is a prestigous competition for school students and consists of six problems to be solved over two days. The questions of the IMO can be found below. [pdf-embedder url="http://gonitsora.com/wp-content/uploads/2016/07/2016_eng.pdf"] If the questions are not...