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Wednesday, April 13, 2022
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The ratio of Alice’s contribution to Bello’s contribution was
This means that for every
If the cost of starting the new business was $9240, then Bello’s
contribution to this cost was
The ratio of Alice’s share of the first year profits to Bello’s
share was
This means that for every
Let the first year profit be
Since Alice’s share of the first year profit was $1881, then
The total profit of the business for the first year was $6897.
The ratio of Alice’s share of the second year profits to Bello’s
share was
This means that for every
Since Bello’s share of the second year profit was $5440, and the profit
that year was $6400, then
Solving this equation, we get
Line
Since
Since
Line
Thus,
Since
Setting
Written in terms of
From part (b), the coordinates of
To determine an expression for the area of
Line
Therefore the
If we call the origin
Since
The area of
We begin by recognizing that a number is a perfect square if the
exponent on each prime factor in its prime factorization is even, and
conversely that the prime factorization of every perfect square has an
even exponent on each prime factor.
The prime factorization of 84 is
For the product
If
The prime factorization of 572 is
Since
That is,
The greatest value of the positive integer
Therefore, the greatest possible value of
The prime factorization of 525 000 is
If
Therefore, if
Suppose that the three powers of 10 chosen from the list are
Since
Since
Similarly,
Thus for some odd positive integers
Therefore, the sum of every choice of three different powers of 10 from
the list is not a perfect square.
Solution 1
For each Bauman string of length 5 in which the first and last
letters are both
For such Bauman strings, either the third letter is
Case 1: The third letter is
In this case, the string is of the form
There are 4 choices for the second letter (
Case 2: The third letter is not
In this case, the string is of the form
The second letter must be different than the third letter and must not
be
Similarly, there are 3 choices for the fourth letter, and so there are
In total, the number of Bauman strings of length 5 in which the first
and last letters are both
Solution 2
For each Bauman string of length 5 in which the first and last
letters are both
Case 1: The second and fourth letters are the same.
In this case, the string is of the form
There are 4 choices for the second letter (
The third letter must be different than the second and fourth letters
(which are the same) and so there are 4 choices for the third
letter.
Thus, there are
Case 2: The second and fourth letters are different.
In this case, the string is of the form
There are 4 choices for the second letter (
The third letter must be different than the second and fourth letters
(which are different) and so there are 3 choices for the third
letter.
Thus, there are
In total, the number of Bauman strings of length 5 in which the first
and last letters are both
Solution 1
We may determine the number of Bauman strings of length 6 that
contain more than one
That is, we may subtract the number of Bauman strings that contain 0
For a Bauman string of length 6 (with no restrictions), there are 5
choices for the first letter and 4 choices for each of the remaining
letters, and so there are a total of
For a Bauman string of length 6 that contains 0
Next, we count the number of Bauman strings that contain exactly 1
If the first letter of the string is a
Similarly, if the last letter is a
Thus, there are
If the second letter of the string is a
Thus, there are
Similarly, if the third, fourth or fifth letter in the string is a
In total, there are
Solution 2
A Bauman string of length 6 cannot contain more than 3
We may determine the number of Bauman strings of length 6 that contain
more than one
That is, we may count the number of strings that contain exactly 3
Thus, there are two cases to consider.
Case 1: The Bauman string has exactly 3
In this case, the string must take one of four possible forms:
There are 4 choices for the second letter (
Similarly, there are
It is worth noting that
We may call such pairs of forms symmetric, and recognize that
under the same restrictions, symmetric forms have an equal number of
Bauman strings.
For strings of the form
Since
Thus, there are
Case 2: The Bauman string has exactly 2
In this case, the string must take one of ten possible forms.
Eight of these occur in one of the following four symmetric pairs
There are 4 choices for the second letter (
Similarly, there are
For strings of the form
Since
Finally, there are
Thus, there are
In total, there are
Solution 1
Consider all Bauman strings of length
There are 4 choices for each of the remaining
Each of these strings either ends with
We call those that end with
Similarly, we call those that do not end in
For example,
Similarly,
We may confirm that
Further, we know that
Next, consider the Bauman strings of length
Each of these strings could have a
Since adding a
From our earlier work, we may confirm that
Further,
Every Bauman string of length
The number of strings
The number of strings
Therefore, we conclude that
From our earlier work, we may confirm that
We use these two formulas
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 | ||
9 | ||
10 | not needed |
Therefore, the number of Bauman strings of length
Solution 2
Let
Further, we define
For example,
Each string in
Therefore,
Similarly, strings that end with
Since we have subtracted the number of strings that begin with
For a Bauman string of length 8, there are 5 choices for the first
letter and 4 choices for each of the remaining 7 letters.
Thus, the total number of Bauman strings of length 8 is equal to
For a Bauman string of length 8 that begins with
Thus, the total number of Bauman strings of length 8 that begin with
Also, the total number of Bauman strings of length 8 that end with
The number of Bauman strings of length 8 that begin with
Therefore, we get
At this point, we could repeat the process above to determine
For integers
The number of such strings,
Thus, the total number of Bauman strings of length
For a Bauman string of length
Thus, the total number of Bauman strings of length
Also, the total number of Bauman strings of length
The number of Bauman strings of length
Therefore, we get