2012 AMC 10B Problems/Problem 20
Problem
Bernardo and Silvia play the following game. An integer between and
, inclusive, is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds
to it and passes the result to Bernardo. The winner is the last person who produces a number less than
. Let
be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of
?
Solution
Let's test each number starting from , for our first case
, we start out with
and the number is then given to Bernardo. He will double the given number so in this case
. So now he gives this resulting number to Silvia and she adds
to the number. So we have
. Now this is then passed to Bernardo so he doubles this
and we have
. And now this is then given to Silvia so
. So she passes it to Bernardo and we have
. Silvia gets it and so we have
. Continuing this we have
. There is no need to continue with this because when Silvia adds