[/quote]
,
be the feet of the altitudes from 
to 
respectively. Let 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a positive integer. The ordered pairs 
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
such that 
and 
are both divisible by 
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
such that 
.
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
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 
.
.
.
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.
Y by
Let 
be an non-isosceles triangle with incenter 
, circumcenter 
and a point 
on segment 
such that 
cut segments 
at
and 
cuts segment 
at 
Circle 
cuts segments 
,again at 
. Let 
The intersection of 
and 
, 
The intersection of 
and 
. Prove that 
are parallel and the median of vertex in triangle 
bisects the arc 
of 
.
be an non-isosceles triangle with incenter 
, circumcenter 
and a point 
on segment 
such that 
cut segments 
at
and 
cuts segment 
at 
Circle 
cuts segments 
,again at 
. Let 
The intersection of 
and 
, 
The intersection of 
and 
. Prove that 
are parallel and the median of vertex in triangle 
bisects the arc 
of 
.This post has been edited 1 time. Last edited by Adam_oly, Sep 20, 2021, 9:09 PM
.
.
is bisector of 
then 
.
.
or 
which means 
is isosceles trapezoid. Therefore 
.
then 
or 
.
then 
lie on a circle. So 
or 
is 
which passes through midpoint of arc 
is cyclic
are parallel
bisects 
, and 
the midpoint of segment 
is cyclic , hence 
, wich implies that the 
.
be the intersection of the tangents to 
at 
, angle chasing gives that 
are collinear , wich is obviously true by 
is a cyclic quadrilateral. 
and by symmetry 
is a cyclic quadrilateral so 
.
,
,
.
is a parallelogram. We know that 
and 
are isogonal conjugates, 
and 
are isogonal conjugates as desired.
