Problem:
Let S be the set of real numbers that can be represented as repeating decimals of the form 0.abc where a,b,c are distinct digits. Find the sum of the elements of S.
Solution:
For any digit x, let x′ denote the digit 9−x. If 0.abc is an element of S, then 0.a′b′c′ is also in S, is not equal to 0.abc
and
0.abc+0.a′b′c′=0.999=1
It follows that S can be partitioned into pairs so that the elements of each pair add to 1. Because S has 10⋅9⋅8=720 elements, the sum of the elements of S is 21​⋅720=360.
OR
Recall that 0.abc=abc/999, so the requested sum is 1/999 times the sum of all numbers of the form abc. To find the sum of their units digits, notice that each of the digits 0 through 9 appears 720/10=72 times, and conclude that their sum is 72(0+1+2+⋯+9)=72⋅45. Similarly, the sum of the tens digits and the sum of the hundreds digits are both equal to 72⋅45. The requested sum is therefore 72⋅45(100+10+1)/999=360.
The problems on this page are the property of the MAA's American Mathematics Competitions