[/quote]
,
be a triangle with centroid 
.
and 
are chosen on rays 
and 
,![\[ \angle ABS=\angle ACR=180^\circ-\angle BGC.\]](http://latex.artofproblemsolving.com/0/a/d/0ad01e29194358ea0d3e10cb74b0769d1bd3a656.png)
Prove that 
.
and 
for which![\[ \left \lfloor \frac{a^2}{b} \right \rfloor + \left \lfloor \frac{b^2}{a} \right \rfloor = \left \lfloor \frac{a^2 + b^2}{ab} \right \rfloor + ab.\]](http://latex.artofproblemsolving.com/c/3/6/c3682397d21443cb720e56251c260b984392a84a.png)
-
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
,
,
,
.
meets 
at 
,
.
be the reflection of 
.
.
meets 
and 
at 
and 
,
meets 
.
,
,
form an arithmetic progression.
such that 
for all 
.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
right and 
,
.
we take two points 
so that 
and 
.
and 
containing inside 
and 
,
and 
,
and 
,
and that 
,
we raise a segment 
perpendicular to the plane of the square. Show that the projections of 
,
,
and 
are coplanar if and only if ![$O\in [AC]\cup [BD]$](http://latex.artofproblemsolving.com/6/6/6/666df8f2afa87f15c79f9c96d9d1cea17a2a7af0.png)
.
.![\[
(2^x - 1)(5^x - 1) = y^n
\]](http://latex.artofproblemsolving.com/f/8/f/f8f013a96863bc5f39aa23edb98eed9aec54a17f.png)
have no positive integer solution 
.
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
