20 pairwise distinct positive integers are all smaller than 70. Prove that among their pairwise differences there are at least 4 equal numbers

1. A supplier of gasoline is considering whether to build a new station. The annual returns will depend on both the size of the station and marketing factors such as the oil industry and local demand of gasoline. The following data is available for decision making:

1. Develop an appropriate decision table for this case.
2. What is the maximax decision?
3. What is the maximin decision?
4. What is the equally likely decision?
5. What is the criterion of realism decision? Use α = 0.8.
6. Develop an opportunity loss table?
7. What is the minimax regret decision?

The function f(t) represents water going into a swimming pool with respect to the number of hours(t) water is flowing in where(t) represents time.f(t)=t squared +8t+9

There is a leak in the pool and it’s losing water at a rate represented by d(t)
d(t)=t squared +11t+4


a. Write a function to represent the amount of water in the pool using the two functions.
b. Use the new function w(t) to determine if the pool will leak all of the water.
c. If the pool will drain of all water, how much time will it take?
d. Will f(t) and d(t) intersect on a graph? Explain what it means if they do.
e. What is the domain of f(t),d(t) , and w(t). Explain your answer.

Prove that for an integer k, k² + 4k + 6 is odd if and only if k is odd.

By using an example, demonstrate the value of the Box-Jenkins Methodobgy. Please make sure to discuss the impact of seasonality.

Need 400 Words

In 75 words describe the Wire Frame and Platonic Solids The main idea with this question is to use the power of the internet to bring some of these geometry lessons to life. Please Type

1 for each of the 5 shapes, can you tell us what the inscribed solid is?

Consider the following sum (which is in expanded form): 1−4 + 7−10 + 13−16 + 19−22 +···±(3n−2).

Note that this is slightly different from the previous sum in that every other term is negative.

(a) Write it as a summation (∑).

(b) Evaluate the sum for every integer n from 1 to 9. (Be careful - if you get this wrong, you will likely get the rest of this question wrong!)

(c) Write a closed-form formula for the value of the sum as a function of n. As in problem 1, do not use a "by cases" or piecewise definition (will need to write a single closed-form expression to receive full credit).(Hint 1: floor and ceiling functions may be useful here.)(Hint 2: try splitting up the sequence of partial sums into two subsequences, finding formulas foreach of the subsequences, then combining the formulas.)

(d) Prove that your formula from part (c) is correct using Mathematical Induction. (You may separateout the cases wherenis even/odd if you wish, but if so please do it as late as possible.)

i. State and prove the Base Case.

ii. State the Inductive Hypothesis.

iii. Show the Inductive Step

Prove that {??+?:?,?∈?} is dense in ? if and only if  r is an irrational number.

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence.

This is all that was given to me... so I am unsure how I am supposed to prove it....

Find the general form of all (integer) solutions for the equation 22x+48y+4z = 18.

Consider the function f(x, y) = 3+xy−x−2y. Let D be the closed triangular region with vertices (1, 4), (5, 0), and (1, 0). Find the absolute maximum and the absolute minimum of f on D.

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if ab>0. Prove that ~ is an equivalence relation. Find the equivalence class [8]. How many distinct equivalence classes are there?

Molson currently sells 41 different brands of beer in Canada. Labatt currently sells 17 different brands.

1. The manager at Mike's Place needs to choose 5 Molson brands and 5 Labatt brands to sell. How many options do they have?
2. The 10 beers (5 Labatt, 5 Molson) selected in the previous question must be placed in a line on a display shelf so that no two Molson products are adjacent and no two Labatt products are adjacent. How many ways are there to do this?
3. Continuing from the previous question, suppose two of the brands selected by the manager were (Labatt) 50 and (Molson) Export. The display shelf can not have a bottle of 50 adjacent to a bottle of Export. How many ways are there to do this while still avoiding two adjacent Molson products and two adjacent Labatt products?
4. How many of the arrangements from the previous question have the bottle of 50 and the bottle of Export among the first (leftmost) 5 bottles?