[/quote]
,
be a trapezoid inscribed in circle 
,
.
is the symmetric point of 
across 
,
intersects 
.
is midpoint of 
is orthocenter of triangle 
.
take a point 
so that 
.
coincide
.
be the point of intersection of 
and let 
.
,
,
such that 
.
and 
be the projections of 
.
meets again 
.
is the point of intersection of 
and 
intersect on the line 
.
by adding every 
?
,
on 
on 
be such that 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
are concurrent.
Prove that![$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$](http://latex.artofproblemsolving.com/8/b/e/8bed6605f3988f59b8395b091b92a8b9e8b40419.png)
Where 
.
intersect at 
be the line through 
the circumcircle of triangle 
.
.
have a common point.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
we define:
A 
between 
,
,
and 
for every 
;
A 
between 
A 
between 
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
.
.
has 
,
,
.
.
Y by
Hello, I was wondering if it's possible to make 8 with the numbers 5, 3, 5, and 7 under the following rules:
-You can only use 5 twice, 3 once, and 7 once.
-You must use all the numbers.
You can stack numbers to form larger numbers (example: I could take 3 and 5 and turn it into 35 or 53, or use 7, 3, and 5 to make 375.)
-You are allowed to use parentheses.
(Also, I already found out that no 3 digital numbers will work for the solution.)
-You can only use 5 twice, 3 once, and 7 once.
-You must use all the numbers.
You can stack numbers to form larger numbers (example: I could take 3 and 5 and turn it into 35 or 53, or use 7, 3, and 5 to make 375.)
-You are allowed to use parentheses.
(Also, I already found out that no 3 digital numbers will work for the solution.)
![\[|P\left(\left\lfloor \sqrt{\sqrt{\left\lceil \sqrt{\left\lfloor \sqrt{\left\lceil \sqrt{\sqrt{\left\lfloor \sqrt{\sqrt{\left\lceil \sqrt{\sqrt{\sqrt{5537}}}\right\rceil !!}}\right\rfloor !}}\right\rceil !}\right\rfloor !}\right\rceil !}}\right\rfloor\right)|\]](http://latex.artofproblemsolving.com/8/7/b/87ba34a74433ea04a72832926fdbc4a618408d4c.png)
![\[
(5 + 5) - (7 - 3) = 8
\]](http://latex.artofproblemsolving.com/e/c/5/ec56494a3dde4ad303e38f5d9e94cb84428e1a51.png)
.
.
.
,
where | | denotes cardinality
,
is equal to 8, so he solved the problem, he is not using eight
