Question

Bernardo and Silvia play the following game. An integer between $0$ and $999$ 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 $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000.$

Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N?$

$7$

$8$

$9$

$10$

$11$

Answer: A

Concepts:recursion inequality digits

Difficulty rating: 1930

Solution:

If the initial number is $x,$ Bernardo's successive outputs are $\begin{gathered} 2x,\quad4x+100,\quad8x+300,\\ 16x+700,\quad32x+1500. \end{gathered}$ Silvia's output after each of these is $50$ larger. Bernardo wins on a given turn exactly when his output is below $1000$ but the following output from Silvia is at least $1000.$

For the first four Bernardo turns, the smallest integer $x$ satisfying those two inequalities is, respectively, $475,213,82,$ and $16.$ A fifth Bernardo output is already at least $1500.$ Thus the smallest winning initial number is $N=16,$ whose digit sum is $1+6=7.$

Thus, the correct answer is A .

2012 AMC 10B Problem 20

Answer: A

Difficulty rating: 1930

Solution:

Problem 20 in Other Years