[/quote]
,
-
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
:o
so 
and since 
is just 
therefore 
so 
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
for which: Every 
be a triangle with incentre 
and circumcentre 
.
be the touchpoints of the incircle with 
,
,
respectively. Prove that 
is the Euler line of 
.
of integers that has the following property:
(possibly 
)
for every integer 
of nonzero integers such that the only admissible set containing both 
and 
satisfies 
,
is the foot from 
，
is a point on segment 
.
meets 
at point 
,
is 
.
is tangent to 
.
be a non-cyclic convex quadrilateral. The feet of perpendiculars from 
are 
respectively, where 
lie on segments 
and 
lies on 
extended. The feet of perpendiculars from 
are 
respectively, where 
lie on segments 
and 
lies on 
extended. Let the orthocenter of 
be 
and 
bisects 
.
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x^2 + y) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/c/7/c/c7cd7c9a4d4372afc23e4503bb1c05fc7d07e788.png)
and let 
respectively. A line through 
and 
while 
and 
and 
are tangent to each other.
of natural numbers such that
is rotated 
degrees about one side. The space through which the hexagon travels forms a solid. Find the volume of this solid.
and height 
:
Volume of cone frustum with base radii 
,
such that: 
.
Y by Adventure10, Mango247
I was working in my math textbook(not the AoPS one) when I came across this math problem:
Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)
I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?
Determine if the events are dependent or independent.
1. Drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles
2.(omitted)
I thought it was independent, since the events happen at the same time, but the textbook answer said dependent.
Can someone help me understand(or prove the textbook wrong)?
This post has been edited 1 time. Last edited by RYang2, Mar 14, 2018, 5:10 PM
Reason: Too general subject
Reason: Too general subject
