2003 AMC 12B Problems

Let $n$ be a 5-digit number, and let q and r be the quotient and remainder, respectively, when $n$ is divided by 100. For how many values of $n$ is $q + r$ divisible by 11?

$\text {(A) } 8180 \qquad \text {(B) } 8181 \qquad \text {(C) } 8182 \qquad \text {(D) } 9000 \qquad \text {(E) } 9090$

Solution

Problem 19

Let $S$ be the set of permutations of the sequence $1,2,3,4,5$ for which the first term is not $1$ . A permutation is chosen randomly from $S$ . The probability that the second term is $2$ , in lowest terms, is $a/b$ . What is $a+b$ ?

$\mathrm{(A)}\ 5\qquad\mathrm{(B)}\ 6\qquad\mathrm{(C)}\ 11\qquad\mathrm{(D)}\ 16\qquad\mathrm{(E)}\ 19$

Solution

Problem 20

Solution

Problem 21

An object moves $8$ cm in a straight line from $A$ to $B$ , turns at an angle $\alpha$ , measured in radians and chosen at random from the interval $(0,\pi)$ , and moves $5$ cm in a straight line to $C$ . What is the probability that $AC < 7$ ?

$\mathrm{(A)}\ 448\qquad\mathrm{(B)}\ 486\qquad\mathrm{(C)}\ 1560\qquad\mathrm{(D)}\ 2001\qquad\mathrm{(E)}\ 2003$

Solution

Problem 22

Problem

Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30$ . Let $N$ be a point on $\overline{AB}$ , and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD}$ , respectively. Which of the following is closest to the minimum possible value of $PQ$ ?