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Difference between revisions of "2001 AMC 12 Problems"

(→‎Problem 5)
(→‎Problem 6)
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== Problem 6 ==
 
== Problem 6 ==
 +
A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents
 +
a different digit. The digits in each part of the number are in decreasing
 +
order; that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</math>. Furthermore,
 +
<math>D</math>, <math>E</math>, and <math>F</math> are consecutive even digits; <math>G</math>, <math>H</math>, <math>I</math>, and <math>J</math> are consecutive odd
 +
digits; and <math>A + B + C = 9</math>. Find <math>A</math>.
 +
 +
<math>\text{(A)}\ 4\qquad \text{(B)}\ 5\qquad \text{(C)}\ 6\qquad \text{(D)}\ 7\qquad \text{(E)}\ 8</math>
  
 
[[2001 AMC 12 Problems/Problem 6|Solution]]
 
[[2001 AMC 12 Problems/Problem 6|Solution]]

Revision as of 13:27, 16 February 2008

Problem 1

The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

$\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D)} 2S + 6 \qquad \text{(E)}\ 2S + 12$

Solution

Problem 2

Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?

$\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad \text{(E)}\ 9$

Solution

Problem 3

The state income tax where Kristin lives is levied at the rate of $p%$ (Error compiling LaTeX. Unknown error_msg) of the first <dollar/>$28000$ of annual income plus $(p + 2)%$ (Error compiling LaTeX. Unknown error_msg) of any amount above <dollar/>$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)%$ (Error compiling LaTeX. Unknown error_msg) of her annual income. What was her annual income?

$\text{(A)}\,$<dollar/>$28000\qquad \text{(B)}\,$<dollar/>$32000\qquad \text{(C)}\,$<dollar/>$35000\qquad \text{(D)}\,$<dollar/>$42000\qquad \text{(E)}\,$<dollar/>$56000$

Solution

Problem 4

The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum?

$\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36$

Solution

Problem 5

What is the product of all positive odd integers less than 10000?

$\text{(A)}\ \dfrac{10000!}{(5000!)^2}\qquad \text{(B)}\ \dfrac{10000!}{2^{5000}}\qquad \text{(C)}\ \dfrac{9999!}{2^{5000}}\qquad \text{(D)}\ \dfrac{10000!}{2^{5000} \cdot 5000!}\qquad \text{(E)}\ \dfrac{5000!}{2^{5000}}$

Solution

Problem 6

A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find $A$.

$\text{(A)}\ 4\qquad \text{(B)}\ 5\qquad \text{(C)}\ 6\qquad \text{(D)}\ 7\qquad \text{(E)}\ 8$

Solution

Problem 7

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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