Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.4 AIME, harder Prove it United States

Problem:

Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends 1024 dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going into a new shop and spending $1024$ . When she comes out of the last shop she has no money left. What is the minimum possible value of $d$ ?

Solution

Solution:

Work backwards. Before going into the last shop she had $1024$ , before the lottery she had $512$ , then $1536$ , $768$ , $2304$ , $1152$ , and so on. We can easily prove by induction that if she ran out of money after $n$ shops, $0 \leq n \leq 10$ , she must have started with $1024 - 2^{10-n}$ dollars. Therefore $d$ is $1023$ .

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