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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

f\left(x + f\left(y\right)\right) = f\left(x + y\right) + f\left(y\right)

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

Solution :

For any positive real numbers z, we have that

f(x+f(y))+z=f(x+y)+f(y)+z
f(f(x+f(y))+z)=f(f(x+y)+f(y)+z)
f(x+f(y)+z)+f(x+f(y))=f(x+y+f(y)+z)+f(x+y)
f(x+y+z)+f(y)+f(x+y)+f(y)=f(x+2y+z)+f(y)+f(x+y)
f(x+y+z)+f(y)=f(x+2y+z)
f(a)+f(b)=f(a+b)

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