MathDB

7

Part of 2017 BMT Spring

Problems(5)

BMT 2017 Spring - Geometry 7

Source:

12/30/2021
Determine the maximal area triangle such that all of its vertices satisfy x29+y216=1\frac{x^2}{9} + \frac{y^2}{16} = 1.
geometry
2017 BMT Team 7

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1/6/2022
There are 8640086400 seconds in a day, which can be deduced from the conversions between seconds, minutes, hours, and days. However, the leading scientists decide that we should decide on 33 new integers x,yx, y, and zz, such that there are xx seconds in a minute, yy minutes in an hour, and zz hours in a day, such that xyz=86400xyz = 86400 as before, but such that the sum x+y+zx + y + z is minimized. What is the smallest possible value of that sum?
number theory
2017 BMT Individual 7

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1/9/2022
What is the sum of the infinite series 203+179+2027+1781+20243+17729+...\frac{20}{3} +\frac{17}{9} + \frac{20}{27} + \frac{17}{81} + \frac{20}{243} + \frac{17}{729} + ... ?
algebra
2017 BMT Discrete #7

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3/9/2024
A light has been placed on every lattice point (point with integer coordinates) on the (infi nite) 2DD plane. De ne the Chebyshev distance between points (x1,y1)(x_1,y_1) and (x2,y2)(x_2, y_2) to be  max(∣x1−x2∣,∣y1−y2∣)\ max (|x_1 - x_2|, |y_1 -y_2|). Each light is turned on with probability 12d/2\frac{1}{2^{d/2}} , where dd is the Chebyshev distance from that point to the origin. What is expected number of lights that have all their directly adjacent lights turned on? (Adjacent points being points such that ∣x1−x2∣+∣y1−y2∣=1|x_1-x_2|+|y_1- y_2| =1.)
combinatorics
2017 BMT Analysis #7

Source:

3/10/2024
Compute ∑k=1∞(−1)k(2k−1)(2k+1)\sum^{\infty}_{k=1} \frac{(-1)^k}{(2k - 1)(2k + 1)}
algebra