Discrete Mathematics Warmups

Discrete Mathematics Warmups: Level 5 Challenges


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

Consider the set S={B,R,I,L,A,N,T} \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 S\mathbb{S} ?.

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

Find 10000P\lceil 10000P \rceil.

Let a0,a1,,a7a_0, a_1, \cdots, a_7 be any 88 distinct integers. Let PP be the product of their pairwise differences, that is:

P=i<j(aiaj)P = \prod _ {i < j} {(a_i - a_j)}

What is the greatest integer which always divides P?P?

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


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