2014 AMC 12B Problems

Amy, Bob, Charlie, Dorothy, Edd, and Frank each select distinct integers between $2005$ and $2014$ , inclusive. What is the probability that the four integers are the lengths of the sides and diagonals of a cyclic quadrilateral?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{1}{42} \qquad \textbf{(C)}\ \frac{1}{30} \qquad \textbf{(D)}\ \frac{1}{21} \qquad \textbf{(E)}\ \frac{1}{7}$

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Let $S$ be the set of points on the graph of $y = x + \sqrt{x}$ such that $x$ is an integer between $-100$ and $100$ , inclusive. How many distinct line segments with endpoints in $S$ have integer side lengths?