AoPS Academy
let 
be the number whose 
-th digit after the decimal point is the 
-th digit after the decimal point of 
. Show that if is rational then so is 
.
from the set 
of positive integers into itself is defined by the equality ![\[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]](http://latex.artofproblemsolving.com/2/a/5/2a5a20395450d37d0352ab75e9e4875ab7803805.png)
for every two relatively prime 
.
the equation 
has a solution.
such that the equation has a unique solution.
be an even integer and let 
be real numbers satisfying 
.
are built on the sides of the triangle 
to the outside. On the segment 
to the outer side of the triangle 
, an equilateral triangle 
is constructed. Prove that 
is the midpoint of the segment 
.
are built on the sides of the triangle to the outside. On the segment to the outer side of the triangle , an equilateral triangle is constructed. Prove that is the midpoint of the segment .
, 
, 
are collinear using Torrichelli point. In the second part we will be proving that 
using rotations, and are collinear
. Then 
is the Torrichelli point of triangle and is the Torichelli point of triangle 
. So , , are collinear. Analogously, points , and are collinear. This means that points , , all lie on line 
, hence are collinear
. Consider rotation centered at point taking 
to 
, then 
is taken to 
, meaning that the segment 
is taken to . Analogously, rotating around you can get 
. Thus, 
. Note that the same can be done with vertices of triangle proving 
. From this follows, that 
.
