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Hexagon ABCDEF is circumscribed about a circle.
AB = 8, CD = 9, EF = 10, and BC = 7.
Find the value of DE + FA.
A circle of radius 2 rolls around the outside of a square of side 4. Find the length of the path
made by the center of the circle.
In how many distinct ways can the letters in LJUBLJANA be
arranged?
Bonus for double points: Of what country is Ljubljana the
capital?
Three traveling salesmen stop at an inn. There is only one small room left. They are tired and take it anyway. The room is $30, so each of the 3 contributes $10. In the morning the manager arrives and decides to give them a partial refund. He gives the bellboy $5 to give to the salesmen. The bellboy realizes the men don’t expect a refund, so he gives them back only $3 and keeps $2 for himself. The men split the refund, taking $1 each. As each man had originally paid $10, but received $1 back, it ended up costing each man $9. They are happy with this and the bellboy is happy as he has $2 in his pocket.Question: each of the 3 men ended up paying $9. 3X9=27+2 (money in bellboy’s pocket) = 29. We started with $30. What happened to the extra $1?
Step 1: Let a=b. Step 2: Then a2 = ab, Step 3: a2 + a2 = a2 + ab , Step 4: 2a2 = a2 + ab , Step 5: 2a2 - 2ab = a2 + ab -2ab, Step 6: 2a2 - 2ab = a2 - ab Step 7: This can be written as 2(a2 - ab) = 1(a2 – ab) Step 8: and canceling the (a2 – ab) from both sides gives 1=2.
1: -1/1 = 1/-1 2: Taking the square root of both sides: √(-1/1) = √(1/-1) 3: Simplifying: √(-1) / √(1) = √(1) / √(-1) 4: In other words, i/1 = 1/i. 5: Therefore, i / 2 = 1 / (2i), 6: i/2 + 3/(2i) = 1/(2i) + 3/(2i), 7: i (i/2 + 3/(2i) ) = i ( 1/(2i) + 3/(2i) ), 8: (i2)/2 + (3i)/2i = i/(2i) + (3i)/(2i), 9: (-1)/2 + 3/2 = 1/2 + 3/2, 10: and this shows that 1=2.
Step 3 is wrong. The problem is that there is no rule that guarantees √(a/b) = √(a) / √(b), except in the case in which a and b are both positive.
If this surprises you, think about the questionWhy should √(a/b) equal √(a)/√(b) ?
If you were to try to convince someone of this, you'd have to start with the
definition of what a "square root" is: it's a number whose square is the number you started with. So all that has to be
true is that √(a) squared is a, √(b) squared is b, and √(a/b) squared is a/b.
So, when you square √(a/b), you will get a/b, and when you square √(a)/√(b), you will also get a/b. That's all that the
definition of square root tells you.
Now, the only way two numbers x and y can have the same square is if x = ±y.
So, what is true is that √(a/b) = ± √(a)/√(b), but in general
there's no reason it has to be √(a/b) = +√(a)/√(b),
rather than √(a/b) = -√(a)/√(b), unless a and b are both positive, for
then (because by convention we take the positive square root) everything in
the above equation is positive.
In our case, it is true that √(-1/1) = √(-1) / √(1), but √(-1/1) (that is, i)
is -√(1)/√(-1) (that is, -1/i) not +√(1)/√(-1) (that is, 1/i)
The fallacy comes from using the latter instead of the former.