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Prove that

(2 + √2)(3 - 2√2)ⁿ + (2 - √2)(3 + 2√2)ⁿ

is an integer for every positive integer n.

 +-----+------+-----+----+
 | NN  | OUSE | AI  | BL |
 +-----+------+-----+----+
 | MI  |  DO  | EON | SI |
 +-----+------+-----+----+
 | LIC |  MA  | NR  | EP |
 +-----+------+-----+----+
 | NG  |  UB  | NI  | YO |
 +-----+------+-----+----+
 | CA  | CILY | DU  | SE |
 +-----+------+-----+----+
(i) Top Beach Vacation Destination	D•••••••• R••••••• (9, 8)

(ii) European Region	S••••• (6)

(iii) Item of Clothing	b••••• (6)

(iv) Downstairs	d•••••• (7)

(v) Hamburger Ingredient	m••••••••• (10)

• Combine groups of letters to create words.
• Each clue has one word associated with it.
• "N letters" shows how many letters each answer has.

In a game show, there is a certain game in which there are four hidden digits. There are no numbers greater than six among them, and no zeros.

You roll a die and then guess if the first digit is higher or lower than what you rolled. (If the die you rolled is equal to the first digit, you win no matter what you said.) You then roll and guess for each of the other three digits.

If you use the best strategy each time when saying "higher" or "lower", what is the chance you will get all four right and win? (Keep in mind you have no idea what the 4 digit number is.)

Suppose you have two candles of the same height but of different widths. One takes four hours to burn all the way while the other takes 7 hours.

Assuming both the candles burn down at steady rates, how long will it taken before one candle is twice as tall as the other?

Two mathematicians, Rex and Ralph, have an ongoing competition to stump each other. Ralph has just finished building a custom house and invites Rex to dinner.

He tells Rex, "The lot my house sits on is a regular polygon. My house is a matching regular polygon sitting on a circular foundation in the middle of the yard. The closest part of the foundation to the edge of my property is exactly the same as the diameter of the foundation. The house is two stories, built around a central circular atrium with a diameter that is exactly one tenth the longest measurement of my property."

"Sounds like quite the house!" remarked Rex.

"Yes, I've been building it for almost two years. When I was excavating the foundation I found a section of an old irrigation pipe. The pipe exactly bisected the area of the house, with each end of the pipe at an edge of the house. After the house was completed while I was preparing to put up a fence around the edge of the property, I discovered that the length of the pipe I found was exactly the length of one side of the property. I used the pipe as a gate across the side of the property with the driveway and built the fence around the other sides. I built a total of 400 feet of fence."

"So what's the square footage of my house?" Ralph asked.

"Are you counting the space occupied by the walls in the square footage?" asked Rex.

"Yes of course," Ralph replied. "It also includes the stairways, hallways, and basically all of the space inside of the exterior walls."

Rex smiles and says, "Nice try, Ralph! Assuming the information you gave me is correct, it'll take just a minute or two to calculate it."

How did Rex know the square footage, and what was his answer?

Six teachers at a certain school come from six different parts of London served by underground stations on the Northern line: Morden, Lambeth, Tooting Bec, Kennington, Borough and Oval.

We have the following information.

(a) Ms Alonso and the teacher from Morden both teach Physics.
(b) Ms Edwards and the teacher from Lambeth both teach Mathematics.
(c) The teacher from Tooting Bec and Mr Chakraborty both teach Computer Science.
(d) The teacher from Oval is older than Mr Chakraborty.
(e) Mr Burton and the teacher from Morden are the two teachers on duty during Monday lunchtime.
(f) Mr Chakraborty and the teacher from Borough are the two teachers on duty during Tuesday lunchtime.
(g) Mr Burton and Mr Fratelli both drink lots of coffee but coffee makes the teacher from Tooting Bec ill.

What subjects do Mr Burton and Ms Dodgson teach and what station does each of them travel from?

[You should assume that each teacher teaches only one subject]

The points of intersection of the curves with equations 21y = 5x² + 2x – 135 and 21y = (x – 5)(21x² + 26x - 225) all lie on a circle C. Find the center and radius of C.

Determine two distinct real numbers a and b that satisfy this system of equations:
a²-b = 73
b²-a =73