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Printjmc
algebra intermediate
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 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not attempt 300 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 .
Find the largest possible two-day success ratio that Beta could have achieved.
Find the largest possible two-day success ratio that Beta could have achieved.
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
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
Final answer
\frac{349}{500}