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Posted: 10/10/2023 9:10 AM
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p4wn3r
Player (42)
Joined: 12/27/2008
Posts: 873
Location: Germany
Lucky TAS of 2011 Gameboy TAS of 2011
For what it's worth, here's the intended solution. Substitute x = a² and y = b². We have a² + b² = 6 (a² + a)(b²+b) = 33/4 Rearrange the second equation as: ab(1+a)(1+b) = ab(1+a+b+ab) = 33/4 Now multiply both sides by 4: 2ab(2+2(a+b)+2ab) = 33 Notice that a² + b² = (a+b)² - 2ab = 6, so 2ab = (a+b)² - 6. Substitute it to find: ((a+b)²-6)((a+b)²+2(a+b)-4)=33 Now set u=a+b and expand: u⁴ + 2u³ -10u² - 12u - 9 = 0 Use rational root theorem or whatever means to see that u=3 is a root. Then factor it (u-3)(u³+5u²+5u+3) = 0 Notice that since u = a+b > 0, and the second factor is always positive for u>0, then u=3 is the only plausible root. Then you have a + b = 3 a²+b² = (a+b)² - 2ab = 6 => ab = 3/2 So, we find a and b by solving the quadratic z²-3z+3/2 to find a = (3+sqrt(3))/2 and b = (3-sqrt(3))/2 to find x = (3+sqrt(3))²/4 and y= (3-sqrt(3))²/4 So the idea was to recast everything in an equation for the sum, which I chose the values to be very simple to solve.
Posted: 10/11/2023 4:12 PM
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OmnipotentEntity
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Player (36)
Joined: 9/11/2004
Posts: 2623
I just realized that I completely misread the original problem as (x + sqrt(x)) / (y + sqrt(y)), so that explains why I was so off. *thud*
Build a man a fire, warm him for a day, Set a man on fire, warm him for the rest of his life.
Posted: 11/11/2023 8:34 AM
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p4wn3r
Player (42)
Joined: 12/27/2008
Posts: 873
Location: Germany
Lucky TAS of 2011 Gameboy TAS of 2011
Today's question has real world applications. Suppose you are a financial analyst working at an investment fund managing billions of dollars in assets, and you are not a corrupt individual (shoot, I just said it had real world applications, and now I make this unrealistic assumption). Someday, someone comes to you saying they have an insurance fund made up of shitcoins they created themselves. A possible example is if they tweeted the following: Now, assume, just in theory, of course I'm not trying to imply anything here, that when they show you the daily value of this fund, instead of calculating it from the assets there, they just use the following Python code, with a Gaussian of mean 7500 and standard deviation 3000, which is summed to the value each day: The question is: what is the probability that this code will increase the value of the fund each day it runs? Do you think this pattern is reasonable for a fund consisting of an asset whose price fluctuates like this? Please, don't try to connect this problem to current events. This is a purely academic question :D
Posted: 12/14/2023 1:55 AM
(Edited: 12/14/2023 4:35 PM)
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OmnipotentEntity
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Player (36)
Joined: 9/11/2004
Posts: 2623
If you have two standard d12s, the probability distribution of the total value of rolling 2 of them is a triangular pmf. How many ways can you relabel the numbers on the d12s to get back the exact same distribution? (subject to the restriction that the values must be positive integers). Is there a generalizeable way find this do this for rolling n dice with k sides? An example for d6s, is the standard dice both have faces {1, 2, 3, 4, 5, 6}, while the (unique) way of relabeling them such that the analogous situation is true is the first dice is labeled {1, 2, 2, 3, 3, 4} and the second is {1, 3, 4, 5, 6, 8}.
Build a man a fire, warm him for a day, Set a man on fire, warm him for the rest of his life.
Posted: 12/24/2023 2:14 AM
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OmnipotentEntity
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Player (36)
Joined: 9/11/2004
Posts: 2623
Apropos of nothing, anyone know an approximate closed form for: 2.405565034289246 This comes from a numeric solution, a, to minimizing the squared error between (1 + erf(x))/2 and 1/(1 + exp(-a x))
Build a man a fire, warm him for a day, Set a man on fire, warm him for the rest of his life.
Posted: 12/24/2023 2:44 AM
(Edited: 12/24/2023 6:32 PM)
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Denial140
Active player (260)
Joined: 8/18/2013
Posts: 145
Location: location, location!
Arcade TAS of 2023
89/37 is not bad given the size of the denominator, around only 0.00016 off, but I suspect this isn't as close as you were looking for. Of course, this came from continued fractions until hitting the relatively large denominator of 7; 89/37 = [2;2,2,7] You may prefer the much closer 197sqrt(2) - 101 - 35pi - 24e with an error of 5.45*10^-9 and which was found by a brute-force computer search.
Current TAS: [SNES] Jelly Boy [NES] Street Fighter 2010
Posted: 12/28/2023 8:28 PM
(Edited: 12/28/2023 8:29 PM)
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BigBoyAdvance
Joined: 11/24/2023
Posts: 10
What number is 400% smaller than 100? Is it 25 or -300? Or 96?
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