## Arithmetic Mean

The arithmetic mean of a set of nine different positive integers is 123456789.  Each number in the set contains a different number of digits with the greatest value being a nine-digit number.  Find the value of each of the nine numbers.
Source: mathcontest.olemiss.edu 1/26/2009

SOLUTION
Given $\left \{a_1,a_2,\cdots,a_9\right \}$ a set of nine different positive integers, the arithmetic mean of the set is

$\frac{1}{9}\sum_{i=1}^{9} a_i$

Define the set of as follows:
$a_1=1$
$a_2=11$
$a_3=111$
$a_4=1111$
$a_5=11111$
$a_6=111111$
$a_7=1111111$
$a_8=11111111$
$a_9=111111111$
Then,
$\sum_{i=1}^9 a_i=123456789$

Since we want the arithmetic mean to be 123456789, all we have to do is multiply each $a_i$ by 9.

Answer: 9; 99; 999; 9999; 99999; 999999; 9999999; 99999999; 999999999.