Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1982 AHSME Problems/Problem 11"

(→‎Problem 11 Solution)
(Add problem statement)
 
(2 intermediate revisions by one other user not shown)
Line 1: Line 1:
−
== Problem 11 Solution ==
+
== Problem ==
  
−
Since <math>BO</math> and <math>CO</math> are angle bisectors of angles <math>B</math> and <math>C</math> respectively, <math>\angle MBO = \angle OBC</math> and similarly <math>\angle NCO = \angle OCB</math>. Because <math>MN</math> and <math>BC</math> are parallel, <math>\angle OBC = \angle MOB</math> and <math>\angle NOC = \angle OCB</math> by corresponding angles. This relation makes <math>\triangle MOB</math> and <math>\triangle NOC</math> isosceles. This makes <math>MB = MO</math> and <math>NO = NC</math>. Therefore the perimeter of <math>\triangle AMN</math> is <math>12 + 18 = 30</math>.
+
How many integers with four different digits are there between <math>1,000</math> and <math>9,999</math> such that the absolute value of  
 +
the difference between the first digit and the last digit is <math>2</math>?
 +
 
 +
<math>\textbf {(A)}\ 672 \qquad
 +
\textbf {(B)}\ 784 \qquad
 +
\textbf {(C)}\ 840 \qquad
 +
\textbf {(D)}\ 896 \qquad
 +
\textbf {(E)}\ 1008</math>    
 +
 
 +
== Solution ==
 +
 
 +
All of the digits of the numbers must be 0 to 9. If the first and last digits are x and y, we have <math>x-y=2</math> and <math>0<x<9</math> or <math>y-x=2,</math> and <math>0<y<9.</math> Substituting we have <math>0<x+2<9,</math> and <math>0<y+2<9.</math> Thus <math>0<x<7</math> and <math>0<y<7,</math> which yields 16 pairs (x, y) such that the absolute value of the difference between the x and y is <math>2.</math> However, we are not done. If 0 is the last digit ( with the pair (0, 2):) then we won't have a 4 digit number, so our real value is 15. Because our digits are distinct, there are <math>(10-2)(10-3)</math> ways to fill the middle 2 places with digits, thus by the multiplication principles (counting) there are <math>15\cdot56 = \boxed {\left(C\right) 840}</math> numbers that fulfill these circumstances.

Latest revision as of 13:32, 16 July 2024

Problem

How many integers with four different digits are there between $1,000$ and $9,999$ such that the absolute value of the difference between the first digit and the last digit is $2$?

$\textbf {(A)}\ 672 \qquad \textbf {(B)}\ 784 \qquad \textbf {(C)}\ 840 \qquad \textbf {(D)}\ 896 \qquad \textbf {(E)}\ 1008$

Solution

All of the digits of the numbers must be 0 to 9. If the first and last digits are x and y, we have $x-y=2$ and $0<x<9$ or $y-x=2,$ and $0<y<9.$ Substituting we have $0<x+2<9,$ and $0<y+2<9.$ Thus $0<x<7$ and $0<y<7,$ which yields 16 pairs (x, y) such that the absolute value of the difference between the x and y is $2.$ However, we are not done. If 0 is the last digit ( with the pair (0, 2):) then we won't have a 4 digit number, so our real value is 15. Because our digits are distinct, there are $(10-2)(10-3)$ ways to fill the middle 2 places with digits, thus by the multiplication principles (counting) there are $15\cdot56 = \boxed {\left(C\right) 840}$ numbers that fulfill these circumstances.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1982_AHSME_Problems/Problem_11&oldid=222791"