Solution for probability, Problem for rate

I'm sorry for not posting in awhile. I went on a family reunion to Yosemite for a week. It was nice :) Also, I'm not sure I'll be able to post as often as I have before, since school has started and I may be busy with homework.


Thanks everyone for trying out the problem and posting in the comments!


Problem:
At a competition with N players, the number of players given elite status is equal to 2^{1+⌊log₂(N-1)⌋} - N. Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of N?


Solution:
We are given the equation  2^{1+⌊log₂(N-1)⌋} - N for the number of players given elite status and we are also told that 19 players have been given elite status. Thus we can make  2^{1+⌊log₂(N-1)⌋} - N equal to 19.


 2^{1+⌊log₂(N-1)⌋} - N = 19


Now we can simply it a little bit.


2^{1+⌊log₂(N-1)⌋} - N
2¹*2^{⌊log₂(N-1)⌋} - N = 19
2^{⌊log₂(N-1)⌋} - N/2 = 19/2
2^{⌊log₂(N-1)⌋} = (19 + N)/2


Now we can put this equation into log form.


2^{⌊log₂(N-1)⌋} = (19 + N)/2
log₂((19+N)/2) = ⌊log₂(N-1)⌋


Because the ⌊log₂(N-1)⌋ has to be an integer (because ⌊x⌋ is the nearest integer ≤ x) we can say that log₂((19 + N)/2) also has to be an integer, because they are equal to each other. Now we can start plugging in values of N that are greater than 19 (there are at least 19 players because 19 of them were given elite status). From the log equation we have (19 + N)/2 = 2^x. Because the N has to be greater than 19 the lowest value 2^x can be is 32. (Try out 16, you can see that it is too small)


(19 + N)/2 = 32
19 + N = 64
N = 45


We have to find the two lowest values of N so now we make 2^x equal to 64


(19 + N)/2 = 64
19 + N = 128
N = 109


We can see that the 2 lowest values of N are 45 and 109. The sum of those two numbers is 154, thus the answer is 154.


Here's the next problem
Problem:
A canoeist paddled upstream for 2 hours, then downstream for 3. The rate of the current was 2 mph. When she stopped, the canoeist realized she was 20 miles downstream form her starting point. How many hours will it take her to paddle back to her starting point?


Good luck! and remember, no calculators are allowed. :)

Solution for probability, Problem for sum

Thanks everyone for trying out the problem, and congrats to those who got the problem right!

Problem:

A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.

Solution:

First of all, we need to find all the possible diameters of the circle. The question  states the numerical value of the area of the circle is less than the numerical value of the circle's circumference. If that is the case, then the radii will be less than 2 because

πr² < 2πr

πr < 2π

r < 2 

If a pair of standard 6-sided die is rolled once, then the diameter will have to be either 2 or 3, because the lowest value you can roll is a 2, and a 4 won't work since the radii has to be less than 2.

There is only 1 possible way to roll a 2 (1,1), and 2 possible ways to roll a 3 (1,2) (2,1).

There are 3 possible rolls out of 36 (6*6) different rolls. Thus, the answer is 1/12.

Here's the next problem:

Problem:

At a competition with N players, the number of players given elite status is equal to 2^{1+⌊log₂(N-1)⌋} - N. Suppose that 19 players are given elite status. What is the sum of the two smallest possible values of N?

⌊x⌋ is the nearest integer ≤ x

Good luck! and remember, no calculators are allowed. :)