MathDB

9

Part of 2017 BMT Spring

Problems(5)

BMT 2017 Spring - Geometry 9

Source:

12/30/2021
Let △ABC\vartriangle ABC be a triangle. Let DD be the point on BCBC such that DADA is tangent to the circumcircle of ABCABC. Let EE be the point on the circumcircle of ABCABC such that DEDE is tangent to the circumcircle of ABCABC, but E≠AE \ne A. Let FF be the intersection of AEAE and BCBC. Given that BF/FC=4/5BF/F C = 4/5, find the maximum possible value for sin⁡∠ACB\sin \angle ACB/
geometry
right triangle, incircle and circumcircle related 2017 BMT Team 9

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1/5/2022
Let AB=10AB = 10 be a diameter of circle PP. Pick point CC on the circle such that AC=8AC = 8. Let the circle with center OO be the incircle of △ABC\vartriangle ABC. Extend line AOAO to intersect circle PP again at DD. Find the length of BDBD.
geometryright triangle
2017 BMT Individual 9

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1/9/2022
The digits 1,4,91, 4, 9 and 22 are each used exactly once to form some 44-digit number NN. What is the sum of all possible values of NN?
number theory
2017 BMT Discrete #9

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3/9/2024
nn balls are placed independently uniformly at random into nn boxes. One box is selected at random, and is found to contain bb balls. Let ene_n be the expected value of b4b^4. Find lim⁡n→∞en.\lim_{n \to \infty}e_n.
combinatorics
2017 BMT Analysis #9

Source:

3/10/2024
Let ada_d be the number of non-negative integer solutions (a,b)(a, b) to a+b=da + b = d where a≡ba \equiv b (mod nn) for a fixed n∈Z+n \in Z^+. Consider the generating function M(t)=a0+a1t+a2t2+...M(t) = a_0 + a_1t + a_2t^2 + ... Consider P(n)=lim⁡t→1(nM(t)−1(1−t)2).P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right). Then P(n)P(n), n∈Z+n \in Z^+ is a polynomial in nn, so we can extend its domain to include all real numbers while having it remain a polynomial. Find P(0)P(0).
number theoryalgebra