Probability
# Discrete Mathematics Warmups

$P$ picked uniformly at random inside an equilateral triangle, is closer to the centroid than to its sides?

What is the probability (to 3 decimal places) that a pointSuppose 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?