Difference between revisions of "2002 AIME II Problems/Problem 1"

(Problem)
(Solution)
Line 7: Line 7:
  
 
== Solution ==
 
== Solution ==
We count the number of three-letter and three-digit palindromes, then subtract the number of license plates containing both types of palindrome.
+
We express the numbers as <math>x=100a+10b+c</math> and <math>100c+10b+c</math>.  From this, we have <center><math>\begin{eqnarray*}z&=&|100a+10b+c-100c-10b-a|\\&=&|99a-99c|\\&=&99|a-c|\\
 
+
\end{eqnarray*}</math></center>
There are <math>10^3\cdot 26^2</math> letter palindromes, <math>10^2\cdot 26^3</math> digit palindromes, and <math>10^2\cdot26^2</math> palindromes that contain both letters and digits.
+
Because <math>a</math> and <math>c</math> are digits, and <math>a</math> is between 1 and 9, there are 9 possible values.
 
 
Since there are <math>10^3\cdot26^3</math> possible plates, the probability desired is <math>\frac{10^2\cdot26^2(10+26-1)}{10^2\cdot26^2\cdot 260}=\frac{35}{260}=\frac{7}{52}</math>. Thus <math>m+n=059</math>.
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2002|n=II|before=First Question|num-a=2}}
 
{{AIME box|year=2002|n=II|before=First Question|num-a=2}}

Revision as of 15:12, 9 August 2008

Problem

Given that

$\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\
   &(2)& \text{y is the number formed by reversing the digits of x; and}\\
&(3)& z=|x-y|. \end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)

How many distinct values of $z$ are possible?

Solution

We express the numbers as $x=100a+10b+c$ and $100c+10b+c$. From this, we have

$\begin{eqnarray*}z&=&|100a+10b+c-100c-10b-a|\\&=&|99a-99c|\\&=&99|a-c|\\ \end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)

Because $a$ and $c$ are digits, and $a$ is between 1 and 9, there are 9 possible values.

See also

2002 AIME II (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS