MathDB

7

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #7 - Symmetric Equations

Source:

10/11/2020
Let a, b,a,\,b, and cc be real numbers such that a+b+c=1a+1b+1ca+b+c=\frac1{a}+\frac1{b}+\frac1{c} and abc=5abc=5. The value of (a−1b)3+(b−1c)3+(c−1a)3\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^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 7

Source:

12/30/2021
Circle Γ\Gamma has radius 1010, center OO, and diameter AB‾\overline{AB}. Point CC lies on Γ\Gamma such that AC=12AC = 12. Let PP be the circumcenter of △AOC\vartriangle AOC. Line APAP intersects Γ\Gamma at QQ, where QQ is different from AA. Then the value of APAQ\frac{AP}{AQ} can be expressed in the form mn\frac{m}{n}, where m and nn are relatively prime positive integers. Compute m+nm + n.
geometry
2020 BMT Team 7

Source:

1/9/2022
A fair six-sided die is rolled five times. The probability that the five die rolls form an increasing sequence where each value is strictly larger than the one that preceded can be written in the form m/nm/n , where mm and nn are relatively prime positive integers. Compute m+nm + n.
probabilitycombinatorics
2020 BMT Individual 7

Source:

1/9/2022
A square has coordinates at (0,0)(0, 0), (4,0)(4, 0), (0,4)(0, 4), and (4,4)(4, 4). Rohith is interested in circles of radius r r centered at the point (1,2)(1, 2). There is a range of radii a<r<ba < r < b where Rohith’s circle intersects the square at exactly 66 points, where aa and bb are positive real numbers. Then b−ab - a can be written in the form m+nm +\sqrt{n}, where mm and nn are integers. Compute m+nm + n.
geometry
2020 BMT Discrete #7

Source:

3/10/2024
Compute the number of ordered triples of positive integers (a,b,c)(a,b,c) such that a+b+c+ab+bc+ac=abc+1a + b + c + ab + bc + ac = abc + 1.
number theory