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 ____________________________________________________________    There are three cookie bakers, Alice, Bob, and Cindy, in the village. Alice works twice as fast as Bob does, and Bob works twice as fast as Cindy does. During the past 3 hours, the three cookie bakers together made 560 cookies. How many of those cookies were made by Bob?    
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   m is the smallest integer such that 2016m is a perfect square. n is the smallest integer such that 2016/n is a perfect square. What is m/n?    
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   An alloy consists of three metals X, Y, and Z. We know the density of X,Y and Z are 3 g/cm3 , 6 g/cm3 , and 8 g/cm3 , respectively. The density of the alloy, which consists of 6 g of X, 18 g of Y, and t g of Z, is 6.4 g/cm3 . What is t? (Assume mixing the metals will not change their combined volume)   
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   Compute 1 − 3 + 5 − 7 + 9 − 11 + · · · + 97 − 99 + 101   
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   Suppose that ABC4 + 200 = ABC9, where A, B, and C are valid digits in base 4 and 9. What is the largest possible value of A + B + C in base 10?   
 

 Jan 16, 2021
 #1
avatar+116126 
+1

There are three cookie bakers, Alice, Bob, and Cindy, in the village. Alice works twice as fast as Bob does, and Bob works twice as fast as Cindy does. During the past 3 hours, the three cookie bakers together made 560 cookies. How many of those cookies were made by Bob?    

 

 

Call  the number of cookies that  Cindy can make in one hour = N

Call the number that Bob can make in one hour =  2N

And let Alice make  2 (2N)  = 4N

 

In  therr hours they can make

 

3 ( N + 2N + 4N) =   560

 

3( 7N)  = 560

 

21N =  560      divide both sides by 21

 

N =  80/3

 

So  in 3 hours Bob can  make    3(2N)  = 3(2* 80/3)  =  2 * 80  =     160

 

 

cool cool cool

 Jan 16, 2021
 #2
avatar+116126 
+1

 m is the smallest integer such that 2016m is a perfect square. n is the smallest integer such that 2016/n is a perfect square. What is m/n?    

 

Factor 2016  =   2^5 * 3^2 * 7

 

We need to have    m  =   2 * 7   = 14    for   2106m   to be a perfect square

 

And we need   n to  be  same for 2016/n   to be a perfect square

 

So.....m/n =  14/14   = 1

 

cool cool cool  

 Jan 16, 2021
 #3
avatar+116126 
+1

Compute 1 − 3 + 5 − 7 + 9 − 11 + · · · + 97 − 99 + 101   

 

We  will have  ( 101  + 1)/ 2  =  51 terms

 

We can write this in a slightly different manner

 

(101 + 1)  + (-99 - 3) + ( 97 + 5)  + (-95 - 7)  .......   

 

We will  have 50 pairings which will sum to   0 

 

The middle  term   will  be unpaired   with any other term  in this summation

 

And  this term  =   51

 

So.....the sum =   51

 

 

cool cool cool

 Jan 16, 2021
 #4
avatar+116126 
+1

  Suppose that ABC4 + 200 = ABC9, where A, B, and C are valid digits in base 4 and 9. What is the largest possible value of A + B + C in base 10? 

 

We have 

 

A(4)^2  + B(4)   + C  +  200   =   A(9)^2  + B(9)  + C

 

A( (9^2 - 4^2)   +  B(9-4)   -   200    =    0

 

65A  +  5B  -  200   =    0

 

13A  + B  - 40    =    0

 

13A  + B   =  40

 

A =  3      B    =  1

 

As large as C  can  be in  base  4  =    3

 

A + B + C    =  3 + 1 + 3    =   7

 

 

cool cool cool

 Jan 16, 2021

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