[/quote]
,
> 
that satisfy 
,
€ 
be a fixed positive integer. Given a set 
of 
points in the plane such that no three are collinear and no four concyclic, let 
be the number of circles 
that contain 
in their interior, and let ![\[m(S)=a_1+a_2+\cdots + a_n.\]](http://latex.artofproblemsolving.com/4/9/b/49b928b1f20e2d1799a1234c7f25a2bc3d62d0dd.png)
Prove that there exists a positive integer 
depending only on 
such that the points of 
are the vertices of a convex polygon if and only if 
such that 
.
and 
.
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
that satisfy the following property: if 
,
,
,
are all the positive divisors of 
with 
,
divides 
for every 
.
,
:
,
Y by
John has 10 marbles with the colors red, green, or blue, all either transparent or translucent. He arranged them in a circle with the following conditions:
3 marbles right beside each other cannot be the same color
The marble across from any marble (assuming John makes a perfect circle) cannot be the same color
Red cannot be in between 2 blues
Blue cannot be between 2 greens
Green cannot be between two Reds
How many different ways can John organize these marbles? State "Impossible" if you think there is no solution. State "Undefined" if one rule doesn't follow another
What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?
Post your answer down below
I just thought of this question right off the top of my head, and I didn't have time to do it, but I'd love to see your answers!
Edit: I just realized "What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?" Is a bit confusing, what I meant was that if you arrange the marbles 2 by 5 (one on the top), the arrow points to what is across from that marble. Same with the 5 by 2, the arrows point that is across.
3 marbles right beside each other cannot be the same color
The marble across from any marble (assuming John makes a perfect circle) cannot be the same color
Red cannot be in between 2 blues
Blue cannot be between 2 greens
Green cannot be between two Reds
How many different ways can John organize these marbles? State "Impossible" if you think there is no solution. State "Undefined" if one rule doesn't follow another
What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?
Post your answer down below
I just thought of this question right off the top of my head, and I didn't have time to do it, but I'd love to see your answers!
Edit: I just realized "What if John arranged them into two rows with 5 marbles each all right beside each other. What about 5 rows with 2 marbles each?" Is a bit confusing, what I meant was that if you arrange the marbles 2 by 5 (one on the top), the arrow points to what is across from that marble. Same with the 5 by 2, the arrows point that is across.
Attachments:
This post has been edited 1 time. Last edited by ilovebender, Mar 18, 2021, 2:56 AM
Reason: confusing
Reason: confusing
