[/quote]
,
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
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 seumAMTH
Two equal circles 
and 
meet at two different points. The line 
intersects at points 
and at points 
respectively (the order on : 
) . Define circles 
and 
as follows: both and touch internally and externally, both and line , and lie in the different halfplanes relatively to line . Suppose that and touch each other. Prove that 
.
I. Voronovich
and 
meet at two different points. The line 
intersects at points 
and at points 
respectively (the order on : 
) . Define circles 
and 
as follows: both and touch internally and externally, both and line , and lie in the different halfplanes relatively to line . Suppose that and touch each other. Prove that 
.I. Voronovich
This post has been edited 1 time. Last edited by parmenides51, Jun 13, 2020, 4:04 PM
be the intersections of 
and let 
be the midpoint of 
.
.
touching 
are collinear and that 
are the in, ex, in similicenters of the pairs of circles 
By Monge, we have that 
be the reflection of 
and hence lies on their radical axis which is 
.
