[/quote]
,
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
be the incentre of acute-angled triangle 
.
,
at 
,
respectively. Let line 
intersect the circumcircle of the triangle at 
and 
,
lies between 
and 
.
so that for every 
in 
.
such that 
.
and 
intersect at a point 
at 
.
be the circumcircle of acute triangle 
and 
intersect at 
and 
.
and 
and 
.
are concyclic.
the centre of the circle passing through 
,
are difined similarly. Prove that 
,
,
are concurrent.
> 
that satisfy 
,
€ 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
such that 
.
and 
.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
,
divides 
for every 
.
,
:
,
Y by Adventure10, Mathlover_1, OronSH, aidan0626, Blue_banana4, PikaPika999, and 7 other users
Given are 
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
scalene triangles.
.
to be the number of bases of these triangles (we count three bases for each equilateral and one for each nonequilateral isosceles triangle). Clearly 
.
isosceles triangles, yielding that there are at least 
scalene triangles. 
points of the same color and we apply the lemma.
points on the perpendicular bisector). Hence there are at most 
isosceles triangles, so at lesat 
scalene triangles.
points of that color.
isosceles triangles. At the same time there are 
triangles in total, making 130 non-isosceles AKA scalene triangles. 
triangles that can be formed.
to 
is the maximum number of isosceles triangles. Then the minimum number of scalene triangles is 
,
as a base, since in order for that to happen, 
isosceles triangles, thus at least 
scalene triangles formed from those points.
isosceles triangles and total triangles equal 
so at least 
triangles.
points in the plane (no three collinear), there are atleast ![\[\binom{x}{3}-2\binom{x}2\]](http://latex.artofproblemsolving.com/e/5/e/e5ebd5bd3558da44fe6d1d25dbe24441be359efa.png)
scalene triangles formed by it.
points and the number of scalene triangles it form are atleast ![\[\binom{13}3-2\binom{13}2=130\]](http://latex.artofproblemsolving.com/3/e/4/3e4c8ec22face221b0219029db3c7ffd7171a46d.png)
as desired.
scalene triangles with same colored vertices.
