[/quote]
,
and 
and 
are real numbers for which the following inequality is satisfied :![\[\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\]](http://latex.artofproblemsolving.com/5/e/4/5e4b3bdc2d5b71e42ba3e6339845794625080eb5.png)
.
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
be positive real numbers such that 
.
When equality holds?
and 
lie on the sides 
and 
of the square 
respectively, and 
.
intersects 
and 
again at 
and 
,
,
,
and 
are the points of tangency of incircle with the center of 
to 
and 
respectively. Let 
.
such that 
.
is the orthocenter of 
,
bisects 
.
,
be a variable point on the line 
such that 
lies between 
and 
and 
intersect at two distinct points 
and 
Prove that the line 
passes through a point independent of 
with real coefficients such that for all reals 
such that 
we have the following relations![\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]](http://latex.artofproblemsolving.com/4/4/9/4491a53f9c6f96c18e42eb984b8e83566140bfdb.png)
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
