1. There are six people sitting at a circular table. Each person is either tall or short. Let a be the number of people sitting next to at least one tall person, and let b be the number of people sitting next to at least one short person. How many possible ordered pairs (a,b) are there? (For example, (6,0) if all six people are tall, since all six people are sitting next to a tall person, and zero people are sitting next to a short person.)

2. Jeff the fly starts at the point (0,0) in the coordinate plane. At each step, Jeff takes a step to the right, left, up, or down. After 10 steps, how many different points could Jeff end up at?

3. Suppose we choose five points on a circle, and draw line segments between every pair of chosen points. Then we can obtain five intersection points, which are marked in red below.

[asy]

unitsize(2 cm);

pair[] A;

pair P;

int i, j;

A[1] = dir(18);

A[2] = dir(60);

A[3] = dir(124);

A[4] = dir(190);

A[5] = dir(330);

A[6] = A[1];

A[7] = A[2];

A[8] = A[3];

A[9] = A[4];

draw(Circle((0,0),1));

for (i = 1; i <= 5; ++i) {

dot(A[i]);

for (j = i + 1; j <= 5; ++j) {

draw(A[i]--A[j]);

}

}

for (i = 1; i <= 5; ++i) {

P = extension(A[i], A[i + 2], A[i + 1], A[i + 3]);

dot(P,red);

}

[/asy]

If we do the same for ten points on a circle, then what is the maximum number of intersection points we can obtain?

Thanks!

Forumofweb2.0cal Feb 28, 2020