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Find the maximum possible number of intersection points between the diagonals of a 13-gon.

Every two hours, a ferry leaves Froopaloop Island and heads due east to the mainland, taking exactly one hour. Then it leaves the mainland and heads due west back to the island, again taking exactly one hour. Then it repeats

Instead of taking the ferry to Froopaloop Island, Heather prefers to do something very risky: she dons her gorgeous white bikini and swims to the island. (The inhabitants of the island don't know what she looks like with clothes on.)

One day, Heather left the mainland terminal at exactly the same time the ferry left the island terminal. Exactly 50 minutes later, the ferry passed her.

Assuming both Heather and the ferry travel at a constant speed, and ignoring the time the ferry spends at the mainland terminal, how much time passed between the ferry passing Heather in the opposite direction and passing her in the same direction?

Hint: Fifty minutes after leaving the island terminal, the ferry will go in the opposite direction of Heather.

A, B, C, and D represent four different digits that can be combined to yield 24 different four-digit integers.

These 24 integers have the following properties:

• 4 are primes.
• 7 are the products of two different odd primes.
• 1 is the square of a prime.
• 8 are divisible by 2 but not by 4.
• 2 are divisible by 4 but not by 8.
• 1 is divisible by 8 but not by 16
• 1 is divisible by 16.

Determine the values of A, B, C, and D.

As part of a Mission Impossible team, Alex has a vital switch to throw in exactly 31 minutes. Unfortunately, his watch has just stopped. All he has are two lengths of fuses, which burn irregularly, and a supply of matches.

One fuse takes 50 minutes to burn completely when lit from either end, and the other which burns in 24 minutes.

How does Alex use these two fuses to time exactly 31 minutes? (A fuse can not be folded to make a shorter time, but it can be burned from both ends to get half the time.)

Let x,y,z be positive real numbers satisfying x+y+z=xyz. Find the minimum value of

   x+y       y+z       z+x
(------- + ------- + -------)2
  1-xy      1-yz      1-zx

Al, Beth, Carl, and Dawn are sitting around a table at a bar, as Al tries to guess Beth’s age. They all know she is at least 21, or she wouldn’t have been allowed into the bar. Al asks Beth five questions, pausing for contemplation after each question:

1. Is your age a multiple of 17?
2. Is your age a multiple of 3?
3. Is your age a prime number?
4. Are you older than I am? (Beth 
knows Al’s age.)
5. Have you celebrated your 51st
birthday?

At this point, Al announces that he has deduced Beth’s age, but Beth tells him he is wrong. Carl, whose age is a prime number, has been listening to this conversation and is able to correctly deduce Al’s age. From his knowledge of Beth, he surmises that she has not answered all the questions truthfully and guesses that she has alternated correct and incorrect answers. He knows that Beth is older than he is, and although he has guessed correctly how many of Beth’s answers are incorrect, he has assumed the wrong ones. So, when he announces what he has deduced as Beth’s age, Beth tells him he is also wrong.

Finally, Dawn who has also been listening in and is sharper than Carl, guesses correctly which of Beth’s answers are incorrect. Now, knowing that Beth is younger than she is, Dawn is able to correctly announce Beth’s age.

What are the ages of Al, Beth, Carl, and Dawn, and what are Al’s and Carl’s incorrect guesses?

It may help to know that Dawn’s age is divisible by 13 and they all know that their ages are all different.

The door to Prof. Adams laboratory has one of those keypad locks that requires entering five digits to open. Unfortunately, he has a hard time remembering the combination, but he has figured out a way to determine it.

The five digits are all different, and he has observed that the first two digits form a perfect square, while the last two digits form a smaller perfect square. Also, the middle digit is the smallest. If he arranges the five digits to form all possible five-digit integers (leading zeros allowed) and adds all these numbers, the sum is a palindrome, with each of its digits a multiple of three.

What is the combination?

A point P is positioned inside regular hexagon ABCDEF so that CP < AP. Triangles BPC, BPE, and APE have areas 7, 12, and 28, respectively. Find the area of the hexagon.

Let ABCDEFGH be a unit cube where A is directly above E, B is directly above F, C is directly above G, and D is directly above H. Let X, Y, and Z be on AG, BH, and CE, respectively such that XG/XA=3/2, YH/YB=2, and ZE/ZC=3. Let O denote the center of the cube. Then find the surface area of tetrahedron OXYZ.