MathDB

6

Part of 2019 BMT Spring

Problems(5)

One of the twins is a prime (BMT 2019 Discrete #6)

Source:

5/25/2019
Define f(n)=n2+n2 f(n) = \dfrac{n^2 + n}{2} . Compute the number of positive integers n n such that f(n)≤1000 f(n) \leq 1000 and f(n) f(n) is the product of two prime numbers.
number theoryprime numbers
Return of the Circle (BMT 2019 Algebra #6)

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5/5/2019
Find the maximum value of xy \dfrac{x}{y} if x x and y y are real numbers such that x2+y2−8x−6y+20=0 x^2 + y^2 - 8x - 6y + 20 = 0 .
The 3-4-5 homothety bisection (BMT 2019 Geo #6)

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5/12/2019
Let △ABE \triangle ABE be a triangle with AB3=BE4=EA5 \frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5} . Let D≠A D \neq A be on line AE‾ \overline{AE} such that AE=ED AE = ED and D D is closer to E E than to A A . Moreover, let C C be a point such that BCDE BCDE is a parallelogram. Furthermore, let M M be on line CD‾ \overline{CD} such that AM‾ \overline{AM} bisects ∠BAE \angle BAE , and let P P be the intersection of AM‾ \overline{AM} and BE‾ \overline{BE} . Compute the ratio of PM PM to the perimeter of △ABE \triangle ABE .
geometrygeometric transformationhomothety
2019 BMT Team 6

Source:

1/7/2022
At a party, 20192019 people decide to form teams of three. To do so, each turn, every person not on a team points to one other person at random. If three people point to each other (that is, AA points to BB, BB points to CC, and CC points to AA), then they form a team. What is the probability that after 65,53665, 536 turns, exactly one person is not on a team
probabilitycombinatorics
color used to paint a die 2019 BMT Individual 6

Source:

1/9/2022
How many square inches of paint are needed to fully paint a regular 66-sided die with side length 22 inches, except for the 13\frac13-inch diameter circular dots marking 11 through 66 (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.
geometry