ShockFish

Heyyy just wondering if you were here too 

Just a fun fact, if it repeats every one digit, then the fraction version is [digit] / 9. If it repeats every two digits, then it is [digit] / 99. You can see the pattern here.

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ShockFish

One thing that you should know is that any quadrilateral's angles sum to 360. Thus, we can make the equation

(q) + (5q + 15) + (3q - 55) + (4q - 20) = 360

This is just simple algebra that you should be able to do. You can simplify this, and then once you find out what q is, figure out what the largest angle is. 

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ShockFish

We have

(3/5)j = 2/3(m)

j - m = (1/5)b

j = b + 35

Now this is just a three-variable system of equations. You should be able to use system of equations to solve this, and then figure out how much Mark has to give to Bill.

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ShockFish

We can simplify first:

x^2 + (b - 6)x + 14 = 0

Now we can use a bit of brute force here:

(x + p)(x + q) 

pq = 14

p + q = b - 6

(p, q) = (14, 1) (7, 2) (-14, -1) (-7, -2)

So we can plug the values of p and q in to get all the possible values of b:

14 + 1 = b - 6 --> b = 21

7 + 2 = b - 6 --> b = 15

-14 + -1 = b - 6 --> b = -9

-7 + -2 = b - 6 --> b = -3

Thus, all the possible values of b are: 21, 15, -9, -3.

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ShockFish

We can use complementary counting, which is, subtracting the possibilities that don't satisfy the condition from the total amount of possibilities.

For the first die, the probability that two dice don't show the same number is 1. 

For the second die, the probability is 5/6, because one possibility was taken from the first die. 

For the third die, the probability is 4/6 or 2/3 because two possibilities have been taken from the two dice already rolled. 

Therefore, we see that the fourth and fifth have 3/6 (1/2) and 2/6 (1/3) as their probability. 

We multiply these all together and we get

1 * 5/6 * 2/3 * 1/2 * 1/3 = 5/54.

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ShockFish

Let's organize our information. 

_ _ 6

There are 9 * 10 ways to fill in the other two spots. (Because the first one can't be 0)

We check the first ones and we find that 

136

216

256 

works.

Each time the number grows by 40. 

So this is (999 - 136) / 40 = 863 / 40 = 21.575, but then you have to count 136, thus the probability that the number is divisible by 8 is 22/90 = 11/45.

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ShockFish