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Nonzero real numbers a, b, and c are such that

a²+b+c=1/a
b²+c+a=1/b
c²+a+b=1/c

Find (a-b)(b-c)(c-a)

3 8 7 9 1 6 4 1 5
2 1 6 3 8 7 3 6 6
5 9 4 2 4 5 2 8 7
4 6 7 1 4 9 8 2 3
9 2 1 7 5 2 1 7 8
8 7 3 1 3 9 6 8 3
6 9 4 5 6 7 2 7 1
5 1 8 7 2 3 1 6 3
3 7 2 6 1 2 8 2 5

The above 9*9 matrix was not generated randomly.
There was some principle in its creation - for you to discover.

So what is it??

In this year's presidential election of Graphistan, 20 nominees compete for the position of head of state. It is your job to schedule a series of public debates between them on national television. There are some constraints you must consider:

1. We do not want it too crowdy, so at most 10 contenders may appear in the same TV show.

2. It is an accepted custom, that candidates shall not attack anyone who is absent. Therefore, ensure that every pair of candidates appears in at least one show together.

Questions:
a) How can you schedule the events with the minimum number of shows?

b) What if there are 22 candidates and the limit is raised to 11 candidates per show?

There are four parts to this problem, numbered 0 to 3:

#0 You have four pots, each with three balls. All the balls are different. You may think of them as having the numbers 1-12 written on them. You are to remove all 12 balls one at a time. The only rule: you must not remove twice in a row from the same pot. How many different ways are there to remove all 12? (solving w/out computer simulation).

#1 The same as #0, but once, and only once, you must replace a ball, returning it to its pot immediately after the removal. So you draw-out 13 times, each time from a different pot. How many possible ways are there?

#2 This time you must replace two balls, replacing each immediately after its removal and replacing each into its pot of origin. It could happen to be the same ball more than once, but you must always move on to a different pot for the next draw, if not done with all 14 draws. How many ways?

#3 Same as #2 but now 3 mandatory replacements are done at some times in the process, again, never drawing from the same pot twice.

PYHIYP means:
Put your hands in your pockets i.e. use no calculators, no pen & paper; just your eyes and your brain.

Insert the missing number:

113 511 375
267 482 250
267 ??? 482

Provide your explanation.