PUMaC 2012 · 团队赛 · 第 5 题

PUMaC 2012 — Team Round — Problem 5

(3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties: √ • The square has side length 7 2. • The boundary of the square intersects the graph of xy = 0 at at least 3 points.

解析

Problem: (3 digits) Among integers { 1 , 2 , . . . , 10 } , let n be the number of numbers for which the sum of the digits is divisible by 5. What are the first three digits (from the left) of n ? Answer: 199 2012 Solution: Since 10 does not work, we only need to concern ourselves with numbers with 2 2011 2012 digits. Let n = a + a 10 + a 10 + · · · + a 10 . Each digit a , . . . , a can be 0 1 2 2011 0 2011 chosen from { 0 , 1 , . . . , 9 } . We have 10 choices for a , 10 choices for a , . . . , 10 choices for a , and finally, 2 choices for 0 1 2010 a . (Given any choices for the first 2011 digits, there are exactly 2 choices for the last digit 2011 for which the sum is divisible by 5.) However, in the process described above, we might have chosen a = a = · · · = a = 0, 0 1 2011 which is not allowed. 2011 Thus, there are 2 · 10 − 1 numbers that work. The number looks like 199 . . . 99, so the first three digits are 199 . Author: Albert