Joyandwarmfuzzies
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Posts posted by Joyandwarmfuzzies
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Oops... the reference to the Pythagorean Theorem is now irrelevant after making the change that DejMar noted, since 4ac being a square doesn't actually matter. I should have removed that, my mistake.
For the roots to be irrational, the discriminant must not be a square number.
Using a proof by contradiction, assume that the roots can be rational. i.e. assume that the discriminant is a square number, k2
b2 - 4ac = k2.
Note that because b is odd, b2 is odd. 4ac must be even. Therefore k2, and therefore k, must also be odd. Let b = 2m + 1 and k = 2n+1
b2 - 4ac = k2
(2m+1)2 - 4ac = (2n+1)2
4m2 + 4m + 1 - 4ac = 4n2 + 4n + 1
4m2 + 4m - 4ac = 4n2 + 4n
m(m+1) - ac = n(n+1)
Note that m(m+1) is even, ac is odd, and n(n+1) is even. An odd number subtracted from an even number is always odd. We have reached a contradiction. Therefore, the discriminant can never be a squared number, and the roots can never be rational.
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Turns out my note about a = c doesn't matter.
For the roots to be irrational, the discriminant must not be a square number.
Using a proof by contradiction, assume that the roots can be rational. i.e. assume that the discriminant is a square number, k2
b2 - 4ac = k2.
Per the Pythagorean Theorem, 4ac must also be a square. Note that because b is odd, b2 is odd. 4ac must be even. Therefore k2, and therefore k, must also be odd. Let b = 2m + 1 and k = 2n+1
b2 - 4ac = k2
(2m+1)2 - 4ac = (2n+1)2
4m2 + 4m + 1 - 4ac = 4n2 + 4n + 1
4m2 + 4m - 4ac = 4n2 + 4n
m(m+1) - ac = n(n+1)
Note that m(m+1) is even, ac is odd, and n(n+1) is even. An odd number subtracted from an even number is always odd. We have reached a contradiction. Therefore, the discriminant can never be a squared number, and the roots can never be rational.
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For the roots to be irrational, the discriminant must not be a square number.
Using a proof by contradiction, assume that the roots can be rational. i.e. assume that the discriminant is a square number, k2
b2 - 4ac = k2.
b2 = 4ac + k2
Per the Pythagoren Theorem, 4ac must also be a square, which only occurs when a = c.
b2 - 4ac = k2
b2 - 4a2 = k2
Note that because b is odd, b2 is odd. And because 4a2 is even, k2, and therefore k, must also be odd. Let b = 2m + 1 and k = 2n+1
b2 - 4a2 = k2
(2m+1)2 - 4a2 = (2n+1)2
4m2 + 4m + 1 - 4a2 = 4n2 + 4n + 1
4m2 + 4m - 4a2 = 4n2 + 4n
m(m+1) - a2 = n(n+1)
Note that m(m+1) is even, a2 is odd, and n(n+1) is even. An odd number subtracted from an even number is always odd. We have reached a contradiction. Therefore, the discriminant can never be a squared number, and the roots can never be rational.
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I, II, III, VIII, XVIII, XXVIII, XXXVIII, LXXXVIII, CLXXXVIII, CCLXXXVIII, CCCLXXXVIII, DCCCLXXXVIII, MDCCCLXXXVIII, etc.
So, it becomes 1, 2, 3, 8, 18, 28, 38, 88, 188, 288, 388, 888, 1888, etc. -
You must have 9 dimes and 10 pennies, giving you a probability of pulling a dime as 9 / 19 = 47.36...%
I must have 5 dimes and 10 nickels, giving me a probability of pulling a dime as 5 / 15 = 33.33...%
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Assuming an equal chance of landing face up and landing face down...
P (face down >= 51) = Sum from i = 51 to 100 of (100 C i) * 2^100
Works out to what superprismatic came up with: 0.4602...
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Average Speed = Total distance / Total time
- Swim: 8 miles over 3 hours
- Run: 3 miles over 20 minutes = 1/3 hours
- Walk: 10 minutes * 0.08 miles per minute = 0.8 miles over 1/6 hours
Average speed = Total Distance / Total Time = 11.8 miles / 3.5 hours = 3.371428... mph = 3 13/35 mph
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Did she say "between Thailand and India" or "between India and Thailand"? If the latter, then it may refer to it alphabetically instead of geographically. Or perhaps both.
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The fifth son's name is What.
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2
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t1 + t2 = 100 minutes = 100/60 hours
t1 = x/55
t2 = (x+20)/40
x/55 + (x+20)/40 = 100/60
x = 1540/57
Total distance = x + (x+20) = 4220/57 miles = 74 2/57 miles -
Each time you make a random cut, you'll intersect all of the previous cuts. Some of the pieces may be very small, but there will still be a piece, as the probability of intersecting at a previous intersection is 0.
The number of pieces, therefore, is one more than the sum of 1, 2, 3, 4, ..., N (since zero cuts gives you one piece), which comes out to N (N+1) / 2 + 1
The cuts are random, not necessarily maximizing number of slices. Otherwise you'd be correct.
Oh damn, good point.
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Each time you make a random cut, you'll intersect all of the previous cuts. Some of the pieces may be very small, but there will still be a piece, as the probability of intersecting at a previous intersection is 0.
The number of pieces, therefore, is one more than the sum of 1, 2, 3, 4, ..., N (since zero cuts gives you one piece), which comes out to N (N+1) / 2 + 1
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Let m be the number of minutes since midnight and n be the number of minutes before noon.
m + n = 720 minutes
m = 9n
n = 72 minutes = 1 hour 12 minutes before noon
Answer: 10:48am -
Assuming there's a 90-degree corner, then the widest moat that's possible to cross is slightly smaller than sqrt(5) / 2.
Reasoning:
Lay one board at a 45-degree angle across the outer corner of the moat, and then rest the other board perpendicular to the first such that it lands on the corner on the inside of the moat.
This gives you a triangle with legs 1 and 1/2, which has a hypotenuse (corresponding to the moat width) of sqrt(5) / 2. The actual width has to be a little less than that, since you lose some length of the board in the process of bearing on the ground and on the other board.I'm not really sure how to post a quick MS Paint image to make it clearer...
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1/3 + sqrt(3) / (2*pi)
How big is the ladder
in New Logic/Math Puzzles
Posted
dgreening, I get the same answer as k-man