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Difference between revisions of "2012 AMC 12B Problems/Problem 14"
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<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
 
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 </math>
 
== Solution==
 
== Solution==
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=== Solution 1 ===
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The last number that Bernado says has to be between 950 and 999. Note that 1->2->52->104->154->308->358->716->776 contains 4 doubling actions. Thus, we have <math>x \rightarrow 2x \rightarrow 2x+50 \rightarrow 4x+100 \rightarrow 4x+150 \rightarrow 8x+300 \rightarrow 8x+350 \rightarrow 16x+700</math>.  
 
The last number that Bernado says has to be between 950 and 999. Note that 1->2->52->104->154->308->358->716->776 contains 4 doubling actions. Thus, we have <math>x \rightarrow 2x \rightarrow 2x+50 \rightarrow 4x+100 \rightarrow 4x+150 \rightarrow 8x+300 \rightarrow 8x+350 \rightarrow 16x+700</math>.  
  
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So the starting number is 16, and our answer is <math>1+6=\boxed{7}</math>, which is A.
 
So the starting number is 16, and our answer is <math>1+6=\boxed{7}</math>, which is A.
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=== Solution 2 ===
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Work backwards. The last number Bernardo produces must be in the range <math>[950,1000)</math>. That means that before this, Silvia must produce a number in the range <math>[475,500)</math>. Before this, Bernardo must produce a number in the range <math>[425,450)</math>. Before this, Silvia must produce a number in the range <math>[213,225)</math>. Before this, Bernardo must produce a number in the range <math>[163,175)</math>. Before this, Silvia must produce a number in the range <math>[82,88)</math>. Before this, Bernardo must produce a number in the range <math>[32,38)</math>. Before this, Silvia must produce a number in the range <math>[16,19)</math>. Bernardo could not have added to any number before this to obtain a number in the range <math>[16,19)</math>, hence the minimum <math>N</math> is 16 with the sum of digits being <math>\boxed{7}</math>.

Revision as of 21:26, 28 February 2012

Problem

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 addes 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 $\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11$

Solution

Solution 1

The last number that Bernado says has to be between 950 and 999. Note that 1->2->52->104->154->308->358->716->776 contains 4 doubling actions. Thus, we have $x \rightarrow 2x \rightarrow 2x+50 \rightarrow 4x+100 \rightarrow 4x+150 \rightarrow 8x+300 \rightarrow 8x+350 \rightarrow 16x+700$.

Thus, $950<16x+700<1000$. Then, $16x>250 \implies x \geq 16$. If $x=16$, we have $16x+700=956$. Working backwards from 956,

$956 \rightarrow 478 \rightarrow 428 \rightarrow 214 \rightarrow 164 \rightarrow 82 \rightarrow 32 \rightarrow 16$.

So the starting number is 16, and our answer is $1+6=\boxed{7}$, which is A.

Solution 2

Work backwards. The last number Bernardo produces must be in the range $[950,1000)$. That means that before this, Silvia must produce a number in the range $[475,500)$. Before this, Bernardo must produce a number in the range $[425,450)$. Before this, Silvia must produce a number in the range $[213,225)$. Before this, Bernardo must produce a number in the range $[163,175)$. Before this, Silvia must produce a number in the range $[82,88)$. Before this, Bernardo must produce a number in the range $[32,38)$. Before this, Silvia must produce a number in the range $[16,19)$. Bernardo could not have added to any number before this to obtain a number in the range $[16,19)$, hence the minimum $N$ is 16 with the sum of digits being $\boxed{7}$.

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