2024 AMC 8
Complete problem set with solutions and individual problem pages
On Monday, Taye has . Every day, he either gains or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, days later?
- A.
- B.
- C.
- D.
- E.
Solution 1
How many dollar values could be on the first day? Only dollars. The second day, you can either add dollars, or double, so you can have dollars, or . For each of these values, you have values for each. For dollars, you have dollars or , and for dollars, you have dollars or dollars. Now, you have values for each of these. For dollars, you have dollars or , for dollars, you have dollars or , for dollars, you have dollars or , and for dollars, you have dollars or .
On the final day, there are 11, 11, 16, and 16 repeating, leaving you with different values.
Solution 2
Continue as in Solution 1 to get , , or dollars by the 2nd day. The only way to get the same dollar amount occurring twice by branching (multiply by or adding ) from here is if or which both aren't true. Hence our answer is .
