Difference between revisions of "Mock AIME 4 2006-2007 Problems/Problem 1"
m (→Solution: wikify, I believe its 1, 10, 100, etc.) |
m |
||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
| − | Albert starts to make a list, in increasing order, of the positive | + | Albert starts to make a list, in [[increasing sequence | increasing]] order, of the [[positive integer]]s that have a first [[digit]] of 1. He writes <math>1, 10, 11, 12, \ldots</math> but by the 1,000th digit he (finally) realizes that the list would contain an [[infinite]] number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order). |
==Solution== | ==Solution== | ||
| + | It is clear that his list begins with 1 one-digit [[integer]], 10 two-digits integers, and 100 three-digit integers, making a total of <math>321</math> digits. | ||
| − | + | So he needs another <math>1000-321=629</math> digits before he stops. He can accomplish this by writing 169 four-digit numbers for a total of <math>321+4(169)=997</math> digits. The last of these 169 four-digit numbers is 1168, so the next three digits will be <math>116</math>. | |
| − | |||
| − | So he needs another <math>1000-321=629</math> digits before he stops. He can accomplish this by writing | ||
==See also== | ==See also== | ||
*[[Mock AIME 4 2006-2007 Problems/Problem 2| Next Problem]] | *[[Mock AIME 4 2006-2007 Problems/Problem 2| Next Problem]] | ||
*[[Mock AIME 4 2006-2007 Problems]] | *[[Mock AIME 4 2006-2007 Problems]] | ||
| + | |||
| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 22:10, 13 February 2007
Problem
Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes 
but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order).
Solution
It is clear that his list begins with 1 one-digit integer, 10 two-digits integers, and 100 three-digit integers, making a total of 
digits.
So he needs another 
digits before he stops. He can accomplish this by writing 169 four-digit numbers for a total of 
digits. The last of these 169 four-digit numbers is 1168, so the next three digits will be 
.
