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Difference between revisions of "2002 AMC 10B Problems"

(Added Problems 3, 4, 6, and 7)
(LaTeXed multiple choice)
Line 12: Line 12:
  
 
Find <math>(2,4,6)</math>.
 
Find <math>(2,4,6)</math>.
 +
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math>
 
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 24 </math>
  
Line 19: Line 20:
 
The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit
 
The arithmetic mean of the nine numbers in the set <math>\{9,99,999,9999,\ldots,999999999\}</math> is a 9-digit number <math>M</math>, all of whose digits are distinct. The number <math>M</math> does not contain the digit
  
−
(A) 0 (B) 2 (C) 4 (D) 6 (E) 8
+
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8 </math>
  
 
[[2002 AMC 10B Problems/Problem 3|Solution]]
 
[[2002 AMC 10B Problems/Problem 3|Solution]]
Line 31: Line 32:
 
when <math>x=4</math>?
 
when <math>x=4</math>?
  
−
(A) 0 (B) 1 (C) 10 (D) 11 (E) 12
+
<math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 10\qquad \mathrm{(D) \ } 11\qquad \mathrm{(E) \ } 12 </math>
  
 
[[2002 AMC 10B Problems/Problem 4|Solution]]
 
[[2002 AMC 10B Problems/Problem 4|Solution]]
Line 43: Line 44:
 
For how many positive integers n is <math>n^2-3n+2</math> a prime number?
 
For how many positive integers n is <math>n^2-3n+2</math> a prime number?
  
−
(A) none (B) one (C) two (D) more than two, but finitely many (E) infinitely many
+
<math> \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} </math>
  
 
[[2002 AMC 10B Problems/Problem 6|Solution]]
 
[[2002 AMC 10B Problems/Problem 6|Solution]]
Line 51: Line 52:
 
Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true?
 
Let <math>n</math> be a positive integer such that <math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{n}</math> is an integer. Which of the following statements is '''not''' true?
  
−
(A) 2 divides n (B) 3 divides n (C) 6 divides n (D) 7 divides n (E) <math>n>84</math>
+
<math> \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 </math>
  
 
[[2002 AMC 10B Problems/Problem 7|Solution]]
 
[[2002 AMC 10B Problems/Problem 7|Solution]]

Revision as of 00:07, 30 December 2008

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

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

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