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

(A) 0 (B) 2 (C) 4 (D) 6 (E) 8

Solution

Problem 4

What is the value of

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

when $x=4$ ?

(A) 0 (B) 1 (C) 10 (D) 11 (E) 12

Solution

Problem 5

Solution

Problem 6

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

(A) none (B) one (C) two (D) more than two, but finitely many (E) 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?