[/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 acute triangle and let 
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
A double-sided ruler allows you to draw a straight line through two given points, as well as a straight line parallel to this one, at a distance of 
cm. Divide the segment into three equal parts using a double-sided ruler, without using a compass.
cm. Divide the segment into three equal parts using a double-sided ruler, without using a compass.This post has been edited 2 times. Last edited by parmenides51, Oct 17, 2020, 9:11 PM
Reason: corrected year
Reason: corrected year
.
and 
.
.
and 
and let the intersection point of 
and the intersection point of 
.
and 
be 
and 
be 
.
be the intersection point of lines 
and 
be on the segment 
and 
.
,
and 
,
and that implies that 
is the midpoint of 
.
and 
.
and 
,
)
,![[asy]
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draw((xmin, 0*xmin-3)--(xmax, 0*xmax-3), linewidth(1)); /* line */
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draw((1.49,3)--(1,0), linewidth(1));
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draw((4.51,-3)--(-2.51,3), linewidth(1));
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draw((4.51,-3)--(1.49,3), linewidth(1));
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dot((4.51,-3),dotstyle);
label("$C$", (4.59,-2.8), NE * labelscalefactor);
label("$l_1$", (-3.59,-2.82), NE * labelscalefactor);
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dot((-2.51,3),linewidth(4pt) + dotstyle);
label("$E$", (-2.43,3.16), NE * labelscalefactor);
dot((5.49,3),linewidth(4pt) + dotstyle);
label("$F$", (5.57,3.16), NE * labelscalefactor);
dot((2.4966666666666666,1),linewidth(4pt) + dotstyle);
label("$G$", (2.57,1.16), NE * labelscalefactor);
dot((1.49,3),linewidth(4pt) + dotstyle);
label("$H$", (1.57,3.16), NE * labelscalefactor);
dot((3.245,1.5),linewidth(4pt) + dotstyle);
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dot((1.245,1.5),linewidth(4pt) + dotstyle);
label("$J$", (1.33,1.66), NE * labelscalefactor);
dot((2.3333333333333335,0),linewidth(4pt) + dotstyle);
label("$K$", (2.41,0.16), NE * labelscalefactor);
dot((3.6666666666666665,0),linewidth(4pt) + dotstyle);
label("$L$", (3.75,0.16), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/9/9/8/998cf92dc8497182d5c0f1a932f3480ac666223c.png)
