-
Prove that
Where 
inscribed in a circle with center 
.
is any point not on (O). 
intersect 
at 
.
be the Lemoine points of triangle 
respectively. Prove that 
are collinear.
denote the set of positive integers. Find all functions 
such that ![\[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\]](http://latex.artofproblemsolving.com/2/3/1/231c0bb0911db66d1a3d2354da7e5dff3004c47f.png)
for all positive integers 
.
Y by
Dear Mathlinkers,
I reseach a synthetic proof whitout calculation…
1. ABC an acutangle triangle acutangle
2. (O), (I) the circum, in-circles
3. DEF the contact triangle
4. M the midpoint of BC
5. P the foot of the perpendicular to EF through M
6. A+ the midpoint of the arc BAC.
Prove : A+, P and I are collinear.
Sincerely
Jean-Louis
I reseach a synthetic proof whitout calculation…
1. ABC an acutangle triangle acutangle
2. (O), (I) the circum, in-circles
3. DEF the contact triangle
4. M the midpoint of BC
5. P the foot of the perpendicular to EF through M
6. A+ the midpoint of the arc BAC.
Prove : A+, P and I are collinear.
Sincerely
Jean-Louis
, then the Quadrilaterals 
and 
are similar to each other with spiral center as 
. Let 
= 
= 
