Wednesday, November 17, 2021
(in North America and South America)
Thursday, November 18, 2021
(outside of North American and South America)
©2021 University of Waterloo
If the row of 11 squares is built from left to right, the first
square requires 4 toothpicks and each subsequent square requires 3
additional toothpicks. These 3 toothpicks form the top edge, the right
edge and the bottom edge. The left edge is formed by the right edge of
the previous square.
Therefore, 11 squares require
Answer: 34
Using the definition of
Answer:
The slope of
Therefore, the line that is parallel to
The midpoint of
Since the line with equation
Answer:
Throughout this solution, we convert the statements about
probability to fractions of the populations in question.
Since 2000 people are asked about side effects and
Since 240 have severe side effects and
This means that the number of people who were given Medicine B and have
severe side effects is
Medicine A | Medicine B | TOTAL | |
---|---|---|---|
Mild side effects | |||
Severe side effects | 160 | 80 | 240 |
TOTAL |
Since 1000 people were given Medicine A and
This means that the number of people who were given Medicine A that have
mild side effects is
Medicine A | Medicine B | TOTAL | |
---|---|---|---|
Mild side effects | 30 | ||
Severe side effects | 160 | 80 | 240 |
TOTAL | 190 |
Similarly, the number of people who were given Medicine B that have
severe or mild side effects is
Medicine A | Medicine B | TOTAL | |
---|---|---|---|
Mild side effects | 30 | 70 | |
Severe side effects | 160 | 80 | 240 |
TOTAL | 190 | 150 |
Since 30 people with mild side effects were given Medicine A and 70
were given Medicine B, then the probability that a random person with
mild side effects was given Medicine B is
Answer:
Solution 1
Since
Since
Therefore,
Thus,
Thus,
We can check by substitution that each of these values of
Solution 2
Let
Then
Since
As in Solution 1,
Since
These equations give
Factoring, we obtain
Since
We can check by substitution that each of these values of
Answer:
Let the side length of the square base
Let
Note that
Let
Join
Consider
Since
Since
Since
Since
Also,
The volume of the square-based pyramid
Triangular-based pyramid
Thus, its volume equals
Therefore, the volume of solid
Since the volume of
We have not yet used the information that
Since
By symmetry,
Since
Since
Since
Since
From
Since
Answer:
The expression
Using (a) with
Alternatively,
If
This gives
Since
We can verify that
We note that
Since we have written
Consider a Leistra sequence with
In the sequence,
Since
The quotients when 216 is divided by these integers are 18, 12, 9, 8, 6,
respectively.
Since
If
If
If
This means that no Leistra sequence starting with
Therefore, there are four Leistra sequences with
Consider a Leistra sequence with
In the sequence,
Since
Therefore,
The odd positive divisors of
The first few powers of 3 are 3, 9, 27, 81. The only such power between
10 and 50 is
The next term in the sequence is an even positive integer
Using a similar argument,
Since each subsequent term
In this case, this means that the only divisor that we will ever be able
to use in this sequence is
This means that the Leistra sequence continues in the following way:
We can again divide out a factor of
Therefore, with
Consider Leistra sequences with
The divisors that can be used to construct the Leistra sequence are
divisors of
Since
This means that the divisors are of the form
From (b), the only possible divisor of the form
Also, the integers of the form
This means that this Leistra sequence can only use the divisors 27 and
18. We note further that the divisor 18 can be used at most once,
otherwise the remaining quotient (and thus next term) would be
odd.
Suppose that the divisor 18 is not used.
Using a similar argument to (b), this gives the sequence
Thus, the sequence cannot use 27 as its only divisor and must use 18
at least once and at most once (thus exactly once).
Therefore, any Leistra sequence beginning with
If the sequence above is extended by using the divisor 18 to generate a
final term, we obtain
The divisor
Also, since
Therefore, the Leistra sequences starting with
There are 17 ways of arranging these 17 divisors as there are 17 places
to choose to place the 18 and the remaining places are filled with the
27s.
Since there are 17 ways of arranging the divisors, there are 17 Leistra
sequences with
Consider Leistra sequences with
The divisors that can be used to construct the Leistra sequence are
divisors of
Since
From (c), these divisors are 18 and 27 along with any divisors of the
form
Since the integers of the form
Thus, these Leistra sequences are generated using the divisors 27, 18,
12, and 36. To count the number of Leistra sequences, we count the
number of ways of arranging the possible divisors, noting that at most 2
factors of 2 can be included amongst the set of divisors used and at
most 50 factors of 3 can be included amongst the set of divisors
used.
Looking at the number of factors of 2, we can see that either (i) the
divisors are all odd, (ii) one divisor of 12 is used, (iii) one divisor
of 36 is used, (iv) two divisors of 18 are used, or (v) one divisor of
18 is used. Any other combination of the even divisors will not satisfy
the conditions of Leistra sequences.
Case 1: 27 only
Note that
Since
Also, we cannot use 15 or fewer 27s, as the sequence can be continued by
dividing by another 27.
Since
Therefore, there are no sequences in this case.
Case 2: 27 and 12
The divisor 12 can only be used once, otherwise too many factors of 2
would be removed.
Since
Therefore, Leistra sequences can be formed by using exactly 16 divisors
equal to 27 and 1 divisor equal to 12.
Since these divisors can be used in any order, there are 17 such
sequences, as in (c).
Case 3: 27 and 36
The divisor 36 can only be used once, otherwise too many factors of 2
would be removed.
Since
Therefore, Leistra sequences can be formed by using exactly 16 divisors
equal to 27 and 1 divisor equal to 36.
Since these divisors can be used in any order, there are 17 such
sequences, as in (c).
Case 4: 27 and two 18s
Note that 18 cannot be used more than two times and that 18 cannot be
combined with 12 or 36, otherwise the remaining quotient would be either
odd or not an integer.
Since
Therefore, Leistra sequences can be formed by using exactly 15 divisors
equal to 27 and 2 divisors equal to 18.
There are
Therefore, there are 136 such sequences.
Case 5: 27 and one 18
Since
As in Case 2 and Case 3, there are 17 such sequences.
Having considered all possibilities, there are
Using
From this, we get
Thus,
Using
If
Since
Thus,
Since
Since
Therefore,
(We note that the functions
Solution 1
Let
From (a),
Solution 2
Let
Since
Squaring both sides, we obtain
Setting
From (b), setting
We will show that
Since
Suppose that
Since
Also, since
Since
Now, for any real number
Now, we know that
If
If
Since
In particular,
From above,
Therefore, when
This is a contradiction, which means that the assumption that