2014 AMC 12B Problems

Problem 1

Find $1 + 2 + 3 + ... + 99 + 100$ .

$\mathrm {(A) } 4950 \qquad \mathrm {(B) } 5000 \qquad \mathrm {(C) } 5050 \qquad \mathrm {(D) } 5075 \qquad \mathrm {(E) } 5100$

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

A circle with radius $R$ is circumscribed around a square. Another circle with radius $r$ is inscribed in the square. Find $\frac{R}{r}$ .

$\mathrm {(A) } \frac{5}{4} \qquad \mathrm {(B) } \sqrt2 \qquad \mathrm {(C) } \frac{3}{2} \qquad \mathrm {(D) } \sqrt3 \qquad \mathrm {(E) } 2$

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Find the remainder when $100! + 99! + 98! + ... + 3! + 2! + 1!$ is divided by $100$ .

$\mathrm {(A) } 11 \qquad \mathrm {(B) } 33 \qquad \mathrm {(C) } 55 \qquad \mathrm {(D) } 77 \qquad \mathrm {(E) } 99$

Solution

Problem 13

Solution

Problem 14

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?