MathDB

Tie 2

Part of 2020 BMT Fall

Problems(4)

2020 BMT Algebra Tiebreakers 2

Source:

1/6/2022
The polynomial f(x)=x3+rx2+sx+tf(x) = x^3 + rx^2 + sx + t has r,sr, s, and tt as its roots (with multiplicity), where f(1)f(1) is rational and t≠0t \ne 0. Compute ∣f(0)∣|f(0)|.
algebra
BMT 2020 Fall - Geometry Tiebreaker 2

Source:

12/30/2021
Quadrilateral ABCDABCD is cyclic with AB=CD=6AB = CD = 6. Given that AC=BD=8AC = BD = 8 and AD+3=BCAD+3 = BC, the area of ABCDABCD can be written in the form pqr\frac{p\sqrt{q}}{r}, where p,qp, q, and r r are positive integers such that pp and r r are relatively prime and that qq is square-free. Compute p+q+rp + q + r.
geometry
2020 BMT Individual Tiebreaker 2

Source:

1/9/2022
Let η∈[0,1]\eta \in [0, 1] be a relative measure of material absorbence. η\eta values for materials combined together are additive. η\eta for a napkin is 1010 times that of a sheet of paper, and a cardboard roll has η=0.75\eta = 0.75. Justin can create a makeshift cup with η=1\eta = 1 using 5050 napkins and nothing else. How many sheets of paper would he need to add to a cardboard roll to create a makeshift cup with η=1\eta = 1?
combinatoricsalgebra
2020 BMT Discrete Tiebreaker #2

Source:

3/10/2024
On a certain planet, the alien inhabitants are born without any arms, legs, or noses. Every year, on their birthday, each alien randomly grows either an arm, a leg, or a nose, with equal probability for each. After its sixth birthday, the probability that an alien will have at least 22 arms, at least 22 legs, and at least 11 nose on the day is mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
combinatorics