Your firm has a monopoly with its Valet-I (pronounced val-ay-eye) app. It knows what’s in the user’s closet and what’s in their calendar, and uses artificial intelligence to plan their wardrobe, ordering pieces as needed from the hottest brands to complement their unique and personal style.Your only marginal cost is €1 each time it is downloaded on the Apploid Store. You have correctly calculated that €10 maximizes European profits if you have to charge each consumer the same price,and at that price you have 100,000 downloads per period.

The Apploid store is considering replacing the per-download fee with a per-period fee of €100,000 regardless of the number of downloads. If the Apploid store makes this change: • explain the effect, if any, that it would have on your per-period price and quantity? • explain what effect, if any, that is would have on your per-period profits.

Market research tells you that some consumers are more price sensitive to others. Unfortunately, when a customer is looking at your app on the Apploid store, you do not know the customer’s price sensitivity. Explain a possible product or pricing strategy—consistent with 2nd degree price discrimination—that your firm can use to increase profits.

The Apploid Store where consumers purchase your app agree to not only let you know each consumer’s (or potential consumer’s) country of residence, but will also allow you to charge different prices in different European countries. Explain what other information will you need, and how you can use this information to increase European profits.

If m is an integer positive, prove that the sequence {xn}n≥0 defined by the following rule is alternating with periodicity m.

xn = [2m | n] + [2m | m + n]

the topic is combinations and Induction

Find the least number of cuts required which can cut a cube into 60 identical pieces? 

How many small boxes of side 1 cm can be cut off from a cube of side 1 metre.

A cuboidal block of 6cm×9cm×12cm is cut up into an exact number of equal cubes. The least possible number of cubes will be

Three cuboids of dimensions 5cm×6cm×7cm,4cm×7cm×8cm and×2cm×3cm×13cm are melted and a cube is made. Find the edge of the cube.

Three rectangular solid metallic cuboids 20cm×10cm×5cm,12cm×10cm×4cm and 15cm×12cm×5cm are melted together and recast into solid cubes each of side 5cm. How many solid cubes can be made from the metal?

If 1=3

2=3

3=5

4=4

5=4

Then, 6=?