Probability
# Discrete Mathematics Warmups

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 \(P?\)

**do not** include any red squares?