MathDB

10

Part of 2015 BMT Spring

Problems(5)

BMT 2015 Spring - Geometry 10

Source:

12/30/2021
Let ABCABC be a triangle with points E,FE, F on CACA, ABAB, respectively. Circle C1C_1 passes through E,FE, F and is tangent to segment BCBC at DD. Suppose that AE=AF=EF=3AE = AF = EF = 3, BF=1BF = 1, and CE=2CE = 2. What is ED2FD2\frac{ED^2}{F D^2} ?
geometry
2015 BMT Team 10

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1/6/2022
Quadratics g(x)=ax2+bx+cg(x) = ax^2 + bx + c and h(x)=dx2+ex+fh(x) = dx^2 + ex + f are such that the six roots of g,hg,h, and g−hg - h are distinct real numbers (in particular, they are not double roots) forming an arithmetic progression in some order. Determine all possible values of a/da/d.
algebra
2015 BMT Spring Analysis 10

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1/20/2022
Evaluate ∫0π/2ln⁡(4sin⁡x)dx.\int^{\pi/2}_0\ln(4\sin x)dx.
integrationcalculus
2015 BMT Spring Discrete 10

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1/20/2022
A partition of a positive integer nn is a summing n1+…+nk=nn_1+\ldots+n_k=n, where n1≥n2≥…≥nkn_1\ge n_2\ge\ldots\ge n_k. Call a partition perfect if every m≤nm\le n can be represented uniquely as a sum of some subset of the nin_i's. How many perfect partitions are there of n=307n=307?
combinatorics
BMT 2015 Spring - Individual 10

Source:

1/22/2022
We have 1010 boxes of different sizes, each one big enough to contain all the smaller boxes when put side by side. We may nest the boxes however we want (and how deeply we want), as long as we put smaller boxes in larger ones. At the end, all boxes should be directly or indirectly nested in the largest box. How many ways can we nest the boxes?
combinatorics