POOL OF PROBLEMS

Using a pool of problems, 16 tests will be formed.

-Every test should have the same number of problems.
-Any problem should be included in at most 8 tests.
-For every 4 tests, there should be at least 1 problem common to all of them.
-Within a test a single problem cannot be used more than once.
-Some tests may be formed of exactly the same problems.

What can be the minimum number of problems in this pool?

If the problem was asked for 8 tests, any problem to be included in at most 4 tests and at least 1 problem common to every 2 tests then the answer would be 6:

(1-2-6), (1-3-5), (1-3-5), (1-4-6), (2-3-6), (2-4-5), (2-4-5), (3-4-6)

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