AoPS Academy
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
, there exists some positive integer 
that makes the following equation have no integer root 
.
that satisfy the equation
lies inside a triangle 
. Let 
and 
be reflections of the point in the lines 
and 
, respectively. Prove that if the triangle 
is equilateral, then the lines 
and 
intersect in a common point.
is equilateral we conclude that is the first Isodyanamic point of 
.We will prove that 
concur at the Fermat point of i.e at 
, the isogonal conjugate of wrt .For this note that 
is isogonal conjugate of 
wrt 
.We know that 
bisects 
.Hence 
are collinear. So done.
, and also 
, thus 
is congruent to 
, so we have 
is the bisector of 
, similarily we get 
are bisectors of 
respectively. hence we know lines and are concurrent at the incenter of .
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
