2002 AMC 10B Problems

Problem 1

The ratio $\frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is:

$\mathrm{(A) \ } 1/6\qquad \mathrm{(B) \ } 1/3\qquad \mathrm{(C) \ } 1/2\qquad \mathrm{(D) \ } 2/3\qquad \mathrm{(E) \ } 3/2$

Solution

Problem 2

For the nonzero numbers a, b, and c, define

$(a,b,c)=\frac{abc}{a+b+c}$

Find $(2,4,6)$ .

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24$

Solution

Problem 3

The arithmetic mean of the nine numbers in the set $\{9,99,999,9999,\ldots,999999999\}$ is a 9-digit number $M$ , all of whose digits are distinct. The number $M$ does not contain the digit

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8$

Solution

Problem 4

What is the value of

$(3x-2)(4x+1)-(3x-2)4x+1$

when $x=4$ ?

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12$

Solution

Problem 5

Solution

Problem 6

For how many positive integers n is $n^2-3n+2$ a prime number?

$\mathrm{(A) \ } \text{none}\qquad \mathrm{(B) \ } \text{one}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many}$

Solution

Problem 7

Let $n$ be a positive integer such that $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}$ is an integer. Which of the following statements is not true?

$\mathrm{(A) \ } 2\text{ divides }n\qquad \mathrm{(B) \ } 3\text{ divides }n\qquad \mathrm{(C) \ } 6\text{ divides }n\qquad \mathrm{(D) \ } 7\text{ divides }n\qquad \mathrm{(E) \ } n>84$

Solution

Problem 8

Suppose July of year $N$ has five Mondays. Which of the following must occurs five times in the August of year $N$ ? (Note: Both months have $31$ days.)

$\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Thursday} \qquad \textbf{(E)}\ \text{Friday}$

Solution

Problem 9

Using the letters $A$ , $M$ , $O$ , $S$ , and $U$ , we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $USAMO$ occupies position

$\mathrm{(A) \ } 112\qquad \mathrm{(B) \ } 113\qquad \mathrm{(C) \ } 114\qquad \mathrm{(D) \ } 115\qquad \mathrm{(E) \ } 116$

Solution

Problem 10

Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has positive solutions $a$ and $b$ . Then the pair $(a,b)$ is

$\mathrm{(A) \ } (-2,1)\qquad \mathrm{(B) \ } (-1,2)\qquad \mathrm{(C) \ } (1,-2)\qquad \mathrm{(D) \ } (2,-1)\qquad \mathrm{(E) \ } (4,4)$

Solution

Problem 11

The product of three consecutive positive integers is $8$ times their sum. What is the sum of the squares?

$\mathrm{(A) \ } 50\qquad \mathrm{(B) \ } 77\qquad \mathrm{(C) \ } 110\qquad \mathrm{(D) \ } 149\qquad \mathrm{(E) \ } 194$

Solution

Problem 12

For which of the following values of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$ ?

$\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Solution

Problem 13

Find the value(s) of $x$ such that $8xy - 12y + 2x - 3 = 0$ is true for all values of $y$ .

$\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac32 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14$

Solution

Problem 14

The number $25^{64}\cdot 64^{25}$ is the square of a positive integer $N$ . In decimal representation, the sum of the digits of $N$ is

$\mathrm{(A) \ } 7\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 35$

Solution

Problem 15

The positive integers $A$ , $B$ , $A-B$ , and $A+B$ are all prime numbers. The sum of these four primes is

$\mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}$

Solution

Problem 16

For how many integers $n$ is $\frac{n}{20-n}$ the square of an integer?

$\textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 10$

Solution

Problem 17

A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\triangle ADG$ .

$\textbf{(A) } 4 + 2\sqrt2 \qquad \textbf{(B) } 6 + \sqrt2\qquad \textbf{(C) } 4 + 3\sqrt2 \qquad \textbf{(D) } 3 + 4\sqrt2 \qquad \textbf{(E) } 8 + \sqrt2$

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Let $\triangle{XOY}$ be a right-triangle with $m\angle{XOY}=90^\circ$ . Let $M$ and $N$ be the midpoints of the legs $OX$ and $OY$ , respectively. Given $XN=19$ and $YM=22$ , find $XY$ .