Y by Adventure10, Mango247
Ok. I think it would be fun to try to "source" every AMC problem from the 10A and 12A. By "source" I mean, try to find an instance where the problem was used before in almost exactly the same form. Probably we can source about half the problems if we try....
We can make this a marathon if you like...
We can make this a marathon if you like...
and 
,
and 
with 
is
?
,
,
.
.
?
students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was 
.
.
times their difference. What is the ratio of the larger number to the smaller?
,
,
,
are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of 
?
more than the radius of the first. What is the relationship between the heights of the two cylinders?
.
,
for some constant 
.
red marbles, 
and 
with 
satisfy 
.
,
lines. How many different values of 
and 
intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area 
.
?
point for every game it draws. Which of the following is 
a true statement about the list of 
?
as a decimal?
has 
,
,
,
,
,
.
times. Let 
,
.
Anyway, I'm pretty certain this is not the first time it appeared in the circle phrasing, but finding tons of questions isomorphic to "how many binary strings don't have consecutive ones" is not exactly hard.
are integers. What is the sum of all possible values of 
have only integer roots for 
as the constant term, but somehow I can't find it right now.
,
and 
,
,
.
are possible?
and 
are not congruent but have the same area and the same perimeter. The sides of 
,
passes through both foci of, and exactly four points on, the ellipse with equation 
.
.
,
be the number of sequences of length 
and 
is divided by 
"
is 
,
.
?
that can be written as fractions 
where 
.![\[(\cos(a\pi)+i\sin(b\pi))^4\]](http://latex.artofproblemsolving.com/e/c/8/ec8719603f64f25dacf3bda313a411e348d627c8.png)
is a real number?
consists of two circles of radii 
and 
that are externally tangent. For 
,
are ordered according to their points of tangency with the 
consists of the 
circles constructed in this way. Let 
,
its radius. What is ![\[\sum_{C\in S}\dfrac1{\sqrt{r(C)}}?\]](http://latex.artofproblemsolving.com/5/b/e/5be96aa0120717982c09c426fc9c7f557c109a84.png)
![[asy]
import olympiad;
size(350);
defaultpen(linewidth(0.7));
// define a bunch of arrays and starting points
pair[] coord = new pair[65];
int[] trav = {32,16,8,4,2,1};
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);
// draw the big circles and the bottom line
path arc1 = arc(coord[0],coord[0].y,260,360);
path arc2 = arc(coord[64],coord[64].y,175,280);
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75));
draw(arc1^^arc2);
draw((-930,0)--(70^2+73^2+850,0));
// We now apply the findCenter function 63 times to get
// the location of the centers of all 63 constructed circles.
// The complicated array setup ensures that all the circles
// will be taken in the right order
for(int i = 0;i<=5;i=i+1)
{
int skip = trav[i];
for(int k=skip;k<=64 - skip; k = k + 2*skip)
{
pair cent1 = coord[k-skip], cent2 = coord[k+skip];
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);
real shiftx = cent1.x + sqrt(4*r1*rn);
coord[k] = (shiftx,rn);
}
// Draw the remaining 63 circles
}
for(int i=1;i<=63;i=i+1)
{
filldraw(circle(coord[i],coord[i].y),gray(0.75));
}[/asy]](http://latex.artofproblemsolving.com/1/1/9/11999bae3eb2df3369375c128acf2c1551a21a07.png)
![\[\frac{1}{\sqrt{r_{\text{middle}}}}=\frac{1}{\sqrt{r_{\text{left}}}}+\frac{1}{\sqrt{r_{\text{right}}}}\]](http://latex.artofproblemsolving.com/0/f/7/0f7b36eeb55e61f9aefd4846adacd43ff8488ba4.png)
where 
Her estimate is larger than 
Her estimate is smaller than 
Her estimate equals 
Her estimate equals 
Her estimate is 
