MathDB

6

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #6 - Sum with Combinations

Source:

10/11/2020
Given that (nk)=n!k!(n−k)!\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}, the value of ∑n=310(n2)(n3)(n+13)\sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} can be written in the form mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Compute m+nm+n.
Bmtalgebra
BMT 2020 Fall - Geometry 6

Source:

12/30/2021
A tetrahedron has four congruent faces, each of which is a triangle with side lengths 66, 55, and 55. If the volume of the tetrahedron is VV , compute V2V^2 .
geometry
2020 BMT Team 6

Source:

1/9/2022
Jack writes whole numbers starting from 1 1 and skips all numbers that contain either a 22 or 99. What is the 100100th number that Jack writes down?
algebra
2020 BMT Individual 6

Source:

1/9/2022
Haydn picks two different integers between 11 and 100100, inclusive, uniformly at random. The probability that their product is divisible by 44 can be expressed in the form m/nm/n , where mm and nn are relatively prime positive integers. Compute m+nm + n.
number theory
2020 BMT Discrete #6

Source:

3/10/2024
Let NN be the number of non-empty subsets TT of S={1,2,3,4,...,2020}S = \{1,2, 3,4,...,2020\} satisfying max(T)>1000max (T) >1000. Compute the largest integer kk such that 3k3^k divides NN.
combinatorics