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# Discrete Mathematics Warmups

If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities.

Suppose two real numbers \(x\) and \(y\) are chosen, uniformly and at random, from the open interval \((0,1)\). Let \(P\) be the probability that the integer closest to \(\dfrac{x}{y}\) is odd.

Find \(\lceil 10000P \rceil\).

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 \(P?\)

In the above image, how many rectangles are there which **do not** include any red squares?

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