MathDB

6

Part of 2014 BMT Spring

Problems(5)

BMT 2014 Spring - Analysis 6

Source:

1/6/2022
Find f(2)f(2) given that ff is a real-valued function that satisfies the equation 4f(x)+(23)(x2+2)f(x−2x)=x3+1.4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.
fefunctional equationalgebra
BMT 2014 Spring - Geometry 6

Source:

12/29/2021
Square ABCDABCD has side length 55 and arc BDBD with center AA. EE is the midpoint of ABAB and CECE intersects arc BDBD at FF. GG is placed onto BCBC such that FGFG is perpendicular to BCBC. What is the length of FGFG?
geometry
2014 BMT Team 6

Source:

1/6/2022
A train is going up a hill with vertical velocity given as a function of tt by 11−t4\frac{1}{1 - t^4} , where tt is between [0,1)[0, 1). Determine its height as a function of tt.
algebra
BMT 2014 Spring - Discrete 6

Source:

1/6/2022
Pick a 33-digit number abcabc, which contains no 00's. The probability that this is a winning number is 1a⋅1b⋅1c\frac1a\cdot\frac1b\cdot\frac1c. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways:
[*] For the lowest digit nn in the number, he rolls an nn-sided die for each time that the digit appears, and gives the number 00 probability of winning if an nn is rolled. [*] For the largest digit mm in the number, he rolls an mm-sided die once and scales the probability of winning by that die roll.
If you choose optimally, what is the probability that your number is a winning number?
probabilitycombinatorics
BMT 2014 Spring - Individual 6

Source:

1/22/2022
Let mm and nn be integers such that m+nm + n and m−nm - n are prime numbers less than 100100. Find the maximal possible value of mnmn.
number theory