[/quote]
,
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 
.
be an acute triangle and let 
.
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 
.
that satisfy the equation 
.
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](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.
and 
Prove that
Y by
The point 
lies inside 
so that 
is isosceles, and angle is a right angle. Furthermore both 
and 
are isosceles with a right angle at 
, and both are outside . Show that 
is isosceles and right-angled.
lies inside 
so that 
is isosceles, and angle is a right angle. Furthermore both 
and 
are isosceles with a right angle at 
, and both are outside . Show that 
is isosceles and right-angled.
This post has been edited 1 time. Last edited by parmenides51, Aug 15, 2020, 9:35 AM
:
:
,
,
is isosceles. 
:
,
,
.
is parallelogram (then 
,
rotation centered at 
clockwise. Denote 
rotation centered at 
.
is compostion of rotations whose angle sum to 
and denote 
.
is isosceles right triangle, hence 
.
,
,
,
is right isosceles. 
be midpoints of 
;
anticlockwise rotation centered at 
sends 
and 
to 
;
,
is a parallelogram, hence 
and 
.
we infer 
(s.a.s. criterion), consequently 
and 
.
,
we get 
;
,
