Maths Olympiad Prep

Track / Stage 3 / 15 of 260 #15 of 1964

Problem 15

AMC 10/12, early questions
Algebra Difficulty 3.0 Find the answer CEMC Fermat

At the end of the year 2000, Steve had $100\$100 and Wayne had $10000\$10000. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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Official solution

We make a table of the total amount of money that each of Steve and Wayne have at the end of each year. After the year 2000, each entry in Steve's column is found by doubling the previous entry and each entry in Wayne's column is found by dividing the previous entry by 2. We stop when the entry in Steve's column is larger than that in Wayne's column: Year Steve Wayne 2000 100 10000 2001 200 5000 2002 400 2500 2003 800 1250 2004 1600 625\text{Year Steve Wayne 2000 100 10000 2001 200 5000 2002 400 2500 2003 800 1250 2004 1600 625} Therefore, 2004 is the first time at which Steve has more money than Wayne at the end of the year.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.