Pirate Gold Problem

There are 5 rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. In case of a tie vote the proposer has the casting vote. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.

Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal. The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from the main proposal. 

Ones and nines

Show that all the divisors of any number of the form 19...9 (with an odd number of nines) end in 1 or 9.  For example, the numbers 19, 1999, 199999, and 19999999 are prime (so clearly the property holds), and the (positive) divisors of 1999999999 are 1, 31, 64516129 and 1999999999 itself.
Show further that this property continues to hold if we insert an equal number of zeroes before the nines.  For example, the numbers 109, 1000999, 10000099999, 100000009999999, and 1000000000999999999 are prime, and the (positive)divisors of 10000000000099999999999 are 1, 19, 62655709, 1190458471, 8400125030569, 159602375580811, 526315789478947368421, and 10000000000099999999999 itself.

Ant in a field

An ant, located in a square field, is 13 meters from one of the corner posts of the field, 17 meters from the corner post diagonally opposite that one, and 20 meters from a third corner post.  Find the area of the field.  Assume the land is flat.

Missing digits

Given that 37! = 13763753091226345046315979581abcdefgh0000000, determine, with a minimum of arithmetical effort, the digits a, b, c, d, e, f, g, and h.  No calculators or computers allowed!

Five card trick

Two information theoreticians, A and B, perform a trick with a shuffled deck of cards, jokers removed.  A asks a member of the audience to select five cards at random from the deck.  The audience member passes the five cards to A, who examines them, and hands one back.  A then arranges the remaining four cards in some way and places them face down, in a neat pile.
B, who has not witnessed these proceedings, then enters the room, looks at the four cards, and determines the missing fifth card, held by the audience member.  How is this trick done?
Note: The only communication between A and B is via the arrangement of the four cards.  There is no encoded speech or hand signals or ESP, no bent or marked cards, no clue in the orientation of the pile of four cards...