# Blog Archives

## IMO 1983 – Problem 3

Let $N$ and $k$ be positive integers and  let $S$ be a set of  $n$ points in the plane such that

$(i)$ no three points of $S$ are collinear, and
$(ii)$  for any point $P$ of  $S$, there are at least $k$ points of $S$ equidistant from $P$

Prove that  $k$  $<$  $\frac{1}{2}$  $+$  $\sqrt{2n}$

Try the question …
Solution will be updated soon

## IMO 2007 Short list problem

Find all functions  $\large f:\mathbb{R+} \rightarrow \mathbb{R+}$ such that

for all x,y $\large \in$  $\large \mathbb{R+}$

Solution :

For any positive real numbers , we have that

and by Cauchy in positive reals, then $f(x)=\alpha x$ for all $x\in (0,\infty)$
Now it’s easy to see that $\alpha=2$ , then $f(x)=2x$    for all positive real numbers $x$