[/quote]
,
and 
.
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
,
such that the sum of the elements in each row and each column of 
.
.
such that 
.
be an interior point for it such that 
.
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
be positive real integer such that 
Prove that
such that 
and 
Y by mathematicsy, Adventure10
Consider a triangle 
such that 
and 
. Three circles with centers 
and 
are pairwise tangent in points on the three sides of the triangle.
Determine the area of the region enclosed by the three circles (the grey area in the figure).
![[asy]
unitsize(0.2 cm);
pair A, B, C;
real[] r;
A = (6,0);
B = (6,6*sqrt(3));
C = (0,0);
r[1] = 3*sqrt(3) - 3;
r[2] = 3*sqrt(3) + 3;
r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7));
draw(A--B--C--cycle);
draw(Circle(A,r[1]));
draw(Circle(B,r[2]));
draw(Circle(C,r[3]));
dot("$A$", A, SE);
dot("$B$", B, NE);
dot("$C$", C, SW);
[/asy]](//latex.artofproblemsolving.com/c/6/9/c692f5b1247b40bc07870d8389f8b87cb7bf413b.png)
such that 
and 
. Three circles with centers 
and 
are pairwise tangent in points on the three sides of the triangle.Determine the area of the region enclosed by the three circles (the grey area in the figure).
![[asy]
unitsize(0.2 cm);
pair A, B, C;
real[] r;
A = (6,0);
B = (6,6*sqrt(3));
C = (0,0);
r[1] = 3*sqrt(3) - 3;
r[2] = 3*sqrt(3) + 3;
r[3] = 9 - 3*sqrt(3);
fill(arc(A,r[1],180,90)--arc(B,r[2],270,240)--arc(C,r[3],60,0)--cycle, gray(0.7));
draw(A--B--C--cycle);
draw(Circle(A,r[1]));
draw(Circle(B,r[2]));
draw(Circle(C,r[3]));
dot("$A$", A, SE);
dot("$B$", B, NE);
dot("$C$", C, SW);
[/asy]](http://latex.artofproblemsolving.com/c/6/9/c692f5b1247b40bc07870d8389f8b87cb7bf413b.png)
This post has been edited 4 times. Last edited by nsato, Feb 14, 2023, 6:15 PM
.
.
be the centers of the middle, the smallest and the largest circles respectively (acc to the OPs figure).
and 
Solving this system we obtain 
.
area of section of circles.
