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If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.

If we pick 77 numbers randomly from the set \(\{1,2,3,4,...,150\}\), we are guaranteed to have at least \(k\) pairs where the difference between the two numbers is 19. What is the maximum possible value of \(k?\)

Please treat a pair as a combination, not a permutation.

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Let \(a_0, a_1, \cdots, a_7\) be *any* \(8\) distinct integers. Let \(P\) be the product of their pairwise differences, that is:

\[P = \prod _ {i < j} {(a_i - a_j)} \]

What is the **greatest** integer which always divides \(P?\)

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Determine the least positive integer \( n \) for which the following condition holds: No matter how the elements of the set of the first \(n\) positive integers, i.e. \( \{1, 2, \ldots n \}\), are colored in red or blue, there are (not necessarily distinct) integers \( x, y, z\), and \( w \) in a set of the same color such that \( x + y + z = w\).

**Details and assumptions**

The phrase **not necessarily distinct** means that the integers can be repeated. For example, if \(1, 2, 4 \) are all colored red, then we have \( 1 + 1 + 2 = 4 \) which would satisfy the condition.

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