MathDB

6

Part of 2017 BMT Spring

Problems(5)

BMT 2017 Spring - Geometry 6

Source:

12/30/2021
Given a cube with side length 1 1, we perform six cuts as follows: one cut parallel to the xyxy-plane, two cuts parallel to the yzyz-plane, and three cuts parallel to the xzxz-plane, where the cuts are made uniformly independent of each other. What is the expected value of the volume of the largest piece?
geometry
rotating a unit square inside a unit circle 2017 BMT Team 6

Source:

1/5/2022
The center of a square of side length 1 1 is placed uniformly at random inside a circle of radius 1 1. Given that we are allowed to rotate the square about its center, what is the probability that the entire square is contained within the circle for some orientation of the square?
geometrysquareprobabilityrotation
2017 BMT Individual 6

Source:

1/9/2022
For how many numbers nn does 20172017 divided by nn have a remainder of either 11 or 22?
number theory
2017 BMT Discrete #6

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3/9/2024
Let S={1,2,...,6}S =\{1, 2,..., 6\}. How many functions f:S→Sf : S \to S are there such that for all s∈Ss \in S, f5(s)=f(f(f(f(f(s)))))=1?f^5(s) = f(f(f(f(f(s))))) = 1?
combinatorics
2017 BMT Analysis #6

Source:

3/10/2024
Consider the function f(x,y,z)=(x−y+z,y−z+x,z−x+y)f(x, y, z) = (x-y +z,y -z +x, z-x+y) and denote by f(n)(x,y,z)f^{(n)}(x, y,z) the function ff applied nn times to the tuple (x,y,z)(x,y,z). Let r1r_1, r2r_2, r3r_3 be the three roots of the equation x3−4x2+12=0x^3- 4x^2 + 12 = 0 and let g(x)=x3+a2x2+a1x+a0g(x) = x^3 + a_2x^2 + a_1x + a_0 be the cubic polynomial with the tuple f(3)(r1,r2,r3)f^{(3)}(r_1, r_2, r_3) as roots. Find the value of a1a_1.
algebra