MathDB

7

Part of 2019 BMT Spring

Problems(5)

The sum of cubes is a multiple of three (BMT 2019 Algebra #7)

Source:

5/5/2019
Let r1 r_1 , r2 r_2 , r3 r_3 be the (possibly complex) roots of the polynomial x3+ax2+bx+43 x^3 + ax^2 + bx + \dfrac{4}{3} . How many pairs of integers a a , b b exist such that r13+r23+r33=0 r_1^3 + r_2^3 + r_3^3 = 0 ?
algebrafactorizationsum of cubes
Geo probability...with trapezoids? That's new. (BMT 2019 Geo #7)

Source:

5/12/2019
Points A,B,C,D A, B, C, D are vertices of an isosceles trapezoid, with AB‾ \overline{AB} parallel to CD‾ \overline{CD} , AB=1 AB = 1 , CD=2 CD = 2 , and BC=1 BC = 1 . Point E E is chosen uniformly and at random on CD‾ \overline{CD} , and let point F F be the point on CD‾ \overline{CD} such that EC=FD EC = FD . Let G G denote the intersection of AE‾ \overline{AE} and BF‾ \overline{BF} , not necessarily in the trapezoid. What is the probability that ∠AGB>30∘ \angle AGB > 30^\circ ?
geometrytrapezoidprobability
I want to make a joke about flux here, but can't (BMT 2019 Discrete #7)

Source:

5/25/2019
(My problem. :D) Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the flux of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?
2019 BMT Team 7

Source:

1/7/2022
How many distinct ordered pairs of integers (b,m,t)(b, m, t) satisfy the equation b8+m4+t2+1=2019b^8+m^4+t^2+1 = 2019?
number theory
ratio m/M wanted, hexagon inscribed in a equilateral 2019 BMT Individual 7

Source:

1/5/2022
Let △ABC\vartriangle ABC be an equilateral triangle with side length MM such that points E1E_1 and E2E_2 lie on side ABAB, F1F_1 and F2F_2 lie on side BCBC, and G1G1 and G2G2 lie on side ACAC, such that m=AE1‾=BE2‾=BF1‾=CF2‾=CG1‾=AG2‾m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2} and the area of polygon E1E2F1F2G1G2E_1E_2F_1F_2G_1G_2 equals the combined areas of △AE1G2\vartriangle AE_1G_2, △BF1E2\vartriangle BF_1E_2, and △CG1F2\vartriangle CG_1F_2. Find the ratio mM\frac{m}{M}. https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png
geometryratiohexagonareasEquilateral