Tie 3

Part of 2019 BMT Spring

Problems(4)

2019 BMT Algebra Tiebreakers 3

Source:

There are two equilateral triangles with a vertex at

(0, 1)

, with another vertex on the line

y = x + 1

and with the final vertex on the parabola

y = x^2 + 1

. Find the area of the larger of the two triangles.

BMT 2019 Spring - Geometry Tiebreaker 3

Source:

We say that a quadrilateral

Q

is tangential if a circle can be inscribed into it, i.e. there exists a circle

C

that does not meet the vertices of

Q

, such that it meets each edge at exactly one point. Let

N

be the number of ways to choose four distinct integers out of

\{1, . . . , 24\}

so that they form the side lengths of a tangential quadrilateral. Find the largest prime factor of

N

2019 BMT Discrete Tiebreakers 3

Source:

\{a, b, c, d, e, f, g, h\}

be a permutation of

\{1, 2, 3, 4, 5, 6, 7, 8\}

. What is the probability that

\overline{abc} +\overline{def}

2019 BMT Individual Tiebreaker 3

Source:

Ankit, Bill, Charlie, Druv, and Ed are playing a game in which they go around shouting numbers in that order. Ankit starts by shouting the number

1

. Bill adds a number that is a factor of the number of letters in his name to Ankit’s number and shouts the result. Charlie does the same with Bill’s number, and so on (once Ed shouts a number, Ankit does the same procedure to Ed’s number, and the game goes on). What is the sum of all possible numbers that can be the

23