MathDB

4

Part of 2013 BMT Spring

Problems(5)

BMT 2013 Spring - Geometry 4

Source:

12/29/2021
Two cubes AA and BB have different side lengths, such that the volume of cube AA is numerically equal to the surface area of cube BB. If the surface area of cube AA is numerically equal to six times the side length of cube BB, what is the ratio of the surface area of cube AA to the volume of cube BB?
geometry
2013 BMT Team 4

Source:

1/5/2022
Find the sum of all real numbers xx such that x2=5x+6x−3x^2 = 5x + 6\sqrt{x} - 3.
algebra
BMT 2013 Spring - Discrete 4

Source:

1/6/2022
Given f1(x)=2x−2f_1(x)=2x-2 and, for k≥2k\ge2, defined fk(x)=f(fk−1(x))f_k(x)=f(f_{k-1}(x)) to be a real-valued function of xx. Find the remainder when f2013(2012)f_{2013}(2012) is divided by the prime 20112011.
number theoryalgebra
BMT 2013 Spring - Analysis 4

Source:

1/6/2022
Given a complex number zz satisfies Im⁡(z)=z2−z\operatorname{Im}(z)=z^2-z, find all possible values of ∣z∣|z|.
complex numbersalgebra
circle inscribed between semicircle and square 2013 BMT Individual 4

Source:

1/2/2022
Let ABCDABCD be a square with side length 22, and let a semicircle with flat side CDCD be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle?
geometrysemicirclesquare