____________________________________________________________ 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?

__________________________________________________

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?

___________________________________________________

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)

____________________________________________________

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

___________________________________________________

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?

Guest Jan 16, 2021

#1**+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

CPhill Jan 16, 2021

#2**+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

CPhill Jan 16, 2021

#3**+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

CPhill Jan 16, 2021

#4**+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

CPhill Jan 16, 2021