MathDB

1

Part of 2007 BAMO

Problems(1)

2007 BAMO p1 marked stick with measures at a row 2,3,5,1,4

Source:

8/27/2019
A 1515-inch-long stick has four marks on it, dividing it into five segments of length 1,2,3,41,2,3, 4, and 55 inches (although not neccessarily in that order) to make a “ruler.” Here is an example. https://cdn.artofproblemsolving.com/attachments/0/e/065d42b36083453f3586970125bedbc804b8a1.png Using this ruler, you could measure 88 inches (between the marks BB and DD) and 1111 inches (between the end of the ruler at AA and the mark at EE), but there’s no way you could measure 1212 inches. Prove that it is impossible to place the four marks on the stick such that the five segments have length 1,2,3,41,2,3, 4, and 55 inches, and such that every integer distance from 11 inch through 1515 inches could be measured.
combinatorics