Discrete Mathematics Warmups

Discrete Mathematics Warmups: Level 5 Challenges


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?

Consider the set \( \mathbb{S} =\{B,R,I,L,A,N,T\}\). How many ways are there for us to partition the set into any number of non empty disjoint subsets whose Union is \(\mathbb{S} \)?.

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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