[/quote]
,
positive integers 
such that for every index 
,
,
is a multiple of 
.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
feet and area 
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
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
