MathDB

5

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #5 - Functional Equation

Source:

10/11/2020
Let f:R+→R+f:\mathbb{R}^+\to \mathbb{R}^+ be a function such that for all x,y∈R+, f(x)f(y)=f(xy)+f(xy)x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right), where R+\mathbb{R}^+ represents the positive real numbers. Given that f(2)=3f(2)=3, compute the last two digits of f(222020)f\left(2^{2^{2020}}\right).
Bmtalgebrafunctionfunctional equation
BMT 2020 Fall - Geometry 5

Source:

12/30/2021
Let A1=(0,0)A_1 = (0, 0), B1=(1,0)B_1 = (1, 0), C1=(1,1)C_1 = (1, 1), D1=(0,1)D_1 = (0, 1). For all i>1i > 1, we recursively define Ai=12020(Ai−1+2019Bi−1),Bi=12020(Bi−1+2019Ci−1)A_i =\frac{1}{2020} (A_{i-1} + 2019B_{i-1}),B_i =\frac{1}{2020} (B_{i-1} + 2019C_{i-1}) Ci=12020(Ci−1+2019Di−1),Di=12020(Di−1+2019Ai−1)C_i =\frac{1}{2020} (C_{i-1} + 2019D_{i-1}), D_i =\frac{1}{2020} (D_{i-1} + 2019A_{i-1}) where all operations are done coordinate-wise. https://cdn.artofproblemsolving.com/attachments/8/7/9b6161656ed2bc70510286dc8cb75cc5bde6c8.png If [AiBiCiDi][A_iB_iC_iD_i] denotes the area of AiBiCiDiA_iB_iC_iD_i, there are positive integers a,ba, b, and cc such that ∑i=1∞[AiBiCiDi]=a2bc\sum_{i=1}^{\infty}[A_iB_iC_iD_i] = \frac{a^2b}{c}, where bb is square-free and cc is as small as possible. Compute the value of a+b+ca + b + c
geometry
2020 BMT Team 5

Source:

1/9/2022
Call a positive integer prime-simple if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to 100100 are prime-simple?
number theory
cut a right cylinder // to its bases into 9 slices 2020 BMT Individual 5

Source:

1/6/2022
A Yule log is shaped like a right cylinder with height 1010 and diameter 55. Freya cuts it parallel to its bases into 99 right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by aπa\pi. Compute aa.
geometry3D geometrycylinder
2020 BMT Discrete #5

Source:

3/10/2024
Let PP be the probability that the product of 20202020 real numbers chosen independently and uniformly at random from the interval [−1,2][-1, 2] is positive. The value of 2P−12P - 1 can be written in the form (mn)b\left(\frac{m}{n} \right)^b, where mm, nn and bb are positive integers such that mm and nn are relatively prime and bb is as large as possible. Compute m+n+bm + n + b.
number theorycombinatorics