Transcript b - Atlanta Sacred Harp

The
Power of a Point
PowerPoint
(with some other kinds of
problems too)
April 1, 2009
Quadrilateral ABCD is
circumscribed about circle O.
Find the perimeter of ABCD.
Answer to Quadrilateral ABCD:
28
Solve the equation log2x216 =x,
where x is real.
Answer to log2x216:
x=3
If the sum of two complex
numbers is 1 and their product
is 1, what is the sum of their
squares?
Answer to sum of squares:
-1
Hexagon ABCDEF is
circumscribed about a circle.
AB = 8, CD = 9, EF = 10,
and BC = 7.
Find the value of DE + FA.
Answer to Hexagon ABCDEF:
20
Find the shaded area.
Answer to shaded area:
30
If 19C8 (“19 choose 8”)
is 75,582,
what is 19C9?
Answer to 19C9:
92,378
Find the center of the circle with
equation
x2 + y2 - 6x + 4√2 y = 64.
Answer to the center of the
circle: (3, -2√2)
What base 8 number does
1100102 represent?
Answer to base 8 number:
628
AC = 4, CD = 5, radius OD = 18.
Find AB.
Answer to AB:
171
4
Find the product of the roots of
6x2 + 17x – 42 = 0
Answer to product of the roots:
-7
Find the constant term of the
expansion of (x2 – 2/x)6
Answer to the constant term:
240
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.
Answer to the circle rolling
around the square:
16 + 4π
In how many distinct ways can
the letters in LJUBLJANA be
arranged?
Bonus for double points: Of
what country is Ljubljana the
capital?
Answer to the number of ways
45,360
Bonus answer: Slovenia
The End of Ciphering
Find the fallacy in each of the
following April Fools problems.
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
question
Why 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.
In fact, the whole proof really boils
down to the fact that
(-1)(-1) = 1,
so √ (-1 * -1) = 1,
but √(-1) * √(-1) = i2 = -1 (not 1).
The proof tried to claim that these two
were equal (but in a more disguised
way where it was harder to spot the
mistake).