MathDB

9

Part of 2015 BMT Spring

Problems(5)

BMT 2015 Spring - Geometry 9

Source:

12/30/2021
Let ABCABC be a triangle. Suppose that a circle with diameter BCBC intersects segments CACA, ABAB at E,FE, F, respectively. Let DD be the midpoint of BCBC. Suppose that ADAD intersects EFEF at XX. If AB=9AB =\sqrt9, AC=10AC =\sqrt{10}, and BC=11BC =\sqrt{11}, what is EXXF\frac{EX}{XF}?
geometry
largest square inscribed in unit cube 2015 BMT Team 9

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1/2/2022
Find the side length of the largest square that can be inscribed in the unit cube.
geometry3D geometrycubegeometric inequalitysquare
2015 BMT Spring Analysis 9

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1/20/2022
Find lim⁡n→∞1n3(n2−12+n2−22+…+n2−(n−1)2).\lim_{n\to\infty}\frac1{n^3}\left(\sqrt{n^2-1^2}+\sqrt{n^2-2^2}+\ldots+\sqrt{n^2-(n-1)^2}\right).
limitsreal analysis
2015 BMT Spring Discrete 9

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1/20/2022
There exists a unique pair of positive integers k,nk,n such that kk is divisible by 66, and ∑i=1ki2=n2\sum_{i=1}^ki^2=n^2. Find (k,n)(k,n).
number theoryDiophantine equation
BMT 2015 Spring - Individual 9

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1/22/2022
The number 2292^{29} has a 99-digit decimal representation that contains all but one of the 1010 (decimal) digits. Determine which digit is missing
number theory