Bernardo and Silvia play the following game. An integer between 0 and

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 ?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Chethan92 · Answer

Let the integer given to Bernardo be X.
Now, Bernardo will double it to 2X and Siliva adds 50 = 2X+50.
This process is repeated.
Now, 2X+50 becomes 4X+100, after Silivia it becomes, 4X+150
4X+150 = 8X+300, after Silivia = 8X+350.
8X+350 = 16X+700, after silivia = 16X+750.
The next process will result in 32X+1500. But the number has to be less than 1000 and bernardo has to win.
Then we need to stop at 16X+700.
16X+700 < 1000.
16X < 300
X = 18, will give 288 < 300.
Bernardo wins at 16X+700 = 16*18+700 = 988.
18 is the smallest initial number, which results in a win for Bernardo.
Sum of its digits = 1+8 = 9

C is the answer

IanStewart · Answer

Notice though that we're trying to find the smallest possible value of X, not the largest, which is what you did at this step in the solution. It is possible for X to be less than 18. For the game to end at this point, 16X + 700 must not only be less than 1000, but it also must be greater than 950. If you solve 16X + 700 > 950, you find 16X > 250, and X = 16 is the smallest integer value of X.

There's also a long but easy-to-follow way to solve. If Bernardo is going to win, then Sylvia must produce a number 1000 or greater. Since she adds 50, she must receive a number between 950 and 999. But Bernardo doubles his numbers, so he must have received something between 475 and 499. Working backwards in that fashion we have:

Sylvia received a number from 950 to 999
Bernardo received a number between 475 to 499
Sylvia received a number between 425 and 449
Bernardo received a number between 213 and 224
Sylvia received a number between 163 and 174
Bernardo received a number between 82 and 87
Sylvia received a number between 32 and 37
Bernardo received a number between 16 and 18

so Bernardo can start with 16 and win the game, and the sum of the digits is 7.

nick1816 · Answer

We have to maximize the number of turns in order minimize the value of N
1st turn of B- 2N 
1st turn of S- 2N+50
2nd turn of B- 4N+100 
2nd turn of S- 4N+150
3rd turn of B- 8N+300 
3rd turn of S- 8N+350
4th turn of B- 16N+700 
4th turn of S- 16N+750 
We have to stop here, as number will exceed 1000 after next turn B, irrespective of any value of N. 
Now we want B to win, hence 16N+700 must be smaller than 1000 and 16N +750 must exceed 1000. 
16N+700<1000 and 16N +750>1000
From above two equations, 15.625<N<18.25
Smallest value of N possible is 16
sum of digits of Smallest value of N= 1+6=7

ScottTargetTestPrep · Answer

Let the initial number be x. Then Bernardo passes 2x to Silvia, Silvia passes 2x + 50 to Bernardo, Bernardo passes 4x + 100 to Silvia etc. Notice that as the number of turns increases, x will decrease; therefore, we want the game to continue for as many turns as possible. Let’s continue the pattern we obtained above:

x -> 2x -> 2x + 50 -> 4x + 100 -> 4x + 150 -> 8x + 300 -> 8x + 350 -> 16x + 700 -> 16x + 750 -> 32x + 1500

We see that at the last step, the generated number is greater than 1000 no matter what the initial number is; therefore, there is no need to continue. In the previous step, 50 is added to 16x + 700, so this was Silvia’s turn. Therefore, if we want Bernardo to win and if we want x to be as small as possible, 16x + 750 must result in a number greater than 1000.

16x + 750 > 1000

16x > 250

x > 15.625

The smallest integer x that satisfies this is x = 16. Indeed, if we take x = 15, then Bernardo passes 16*15 + 700 = 940 to Silvia and Silvia passes 940 + 50 = 990 to Bernardo, resulting in a win for Silvia. If, on the other hand, x = 16, Bernardo passes 16*16 + 700 = 956 to Silvia and wins the game.

Answer: A

RositaDhe · Answer

B = 999*2 = 19 98  = 9+8 = 1 7  = 7

Ans. A

hahahahah i know i might using incorrect rules (Last Digit of Power but above is  Last Digit of Sum , lmao!)

versewriter · Answer

Here is the solution:

The question is asking to take the result number less than 1000 so we take 998 because it is divisible by 2. So lets take the number backward according to the question.

998
998-50
948/2 = 478

478-50
428/2
214

214-50
164/2
82

82-50
32/2
16

16 is our N when it will make Bernardo win in the game. SO lets add the digits of 16, the answer is 7.
So option A is our answer.

bumpbot · Answer

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