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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. See more

What is the probability (to 3 decimal places) that a point \(P\) picked uniformly at random inside an equilateral triangle, is closer to the centroid than to its sides?

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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 \(\lceil 10000P \rceil\).

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

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