Problem:
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of points. Alpha scored points out of points attempted on the first day, and scored points out of points attempted on the second day. Beta, who did not attempt points on the first day, had a positive integer score on each of the two days, and Beta's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was . The largest possible two-day success ratio that Beta could have achieved is , where and are relatively prime positive integers. What is ?
Solution:
Let Beta's scores be out of on day one and out of on day two, so that , and . Then and , so , and . Beta's two-day success ratio is greatest when is greatest. Let and subtract from both sides of the last inequality to obtain . Because , conclude that , and . When , , so . If , then , but then and so . Notice that when and , then and . Thus Beta's maximum possible two-day success ratio is , so .
Let be the total number of points scored by Beta in the two days. Notice first that , because is of , and Beta's success ratio is less than on each day of the competition. Notice next that is possible, because Beta could score point out of attempted on the first day, and out of attempted on the second day. Thus , and .
Note that Beta's two-day success ratio can be greater than Alpha's while Beta's success ratio is less on each day. This is an example of Simpson's Paradox.
The problems on this page are the property of the MAA's American Mathematics Competitions