Wednesday, April 10, 2019
(in North America and South America)
Thursday, April 11, 2019
(outside of North American and South America)
©2019 University of Waterloo
The rectangle in Figure A has dimensions
Solution 1
We begin by labelling Figure B as shown.
The width of Figure B is
However, the width is also equal to
Since
Similarly,
The perimeter of Figure B is
Solution 2
The answer in Solution 1, 30, is equal to the answer inpart (a). Why?
Consider sliding
Since
Further, since
Since
Therefore, the perimeter of Figure B is 30.
Following the approach in Solution 2 above, the perimeter of Figure C is equal to the perimeter of a rectangle with side lengths
Since the perimeter of Figure C is 56, then
(Alternatively, we could have determined that the two missing lengths in Figure C are each equal to
Each of the segments having lengths 4 or 7 may be “pushed out” to show that the perimeter of Figure D is equal to the perimeter of a square with side length
That is, the perimeter of Figure D is
We require the largest integer
Solving this inequality, we get
Thus, the largest integer
In 10 minutes, there are
If the machine is set to make one cut every 8 seconds, then
Rope is fed into the machine at a constant rate of 2 metres per second.
If the machine is set to make one cut every 3 seconds, then the length of each piece of rope that is cut off is
Solution 1
Rope is fed into the machine at a constant rate of 2 metres per second, which is equivalent to
If each piece of rope that is cut off is 30 m long, then the machine is set to make
Solution 2
Rope is fed into the machine at a constant rate of 2 metres per second.
To cut off a piece of rope 30 m long, the machine must be set to make a cut every
If the machine makes a cut every 15 seconds, then it makes
Solution 3
In part (b), the machine was set to make one cut every 3 seconds, and so the length of each piece of rope that is cut off is 6 metres.
Since rope is fed into the machine at a constant rate, to cut off a piece of rope that is times longer (30 m is 5 times longer than 6 m), the machine must be set to make each cut 5 times more slowly, or every
If the machine makes a cut every 15 seconds, then it makes
Solution 1
If the machine is set to make 16 cuts per minute, or 16 cuts every 60 seconds, it will make one cut every
Since the machine is set to make one cut every 3.75 seconds, then the length of each piece of rope that is cut off is
Solution 2
If the machine is set to make 16 cuts per minute, then every 60 seconds it is set to make 16 cuts.
The rope is fed into the machine at a constant rate of 2 metres per second, and so after 60 seconds,
During these 60 seconds, the 120 metres of rope is cut 16 times, and so the length of each piece of rope that is cut off is
Solution 1
In the list of integers beginning at 1, the 6
Thus, Tania has listed each of the integers from 1 to 29 with the exception of the positive multiples of 5 less than 30:
Therefore, just before Tania leaves out the
Solution 2
Beginning at 1, each group of five integers has one integer that is a multiple of 5.
For example, the first group of five integers,
In Tania’s list of integers, she leaves out the integers that are multiples of 5, and so in every group of five integers, Tania lists four of these integers.
Thus, just before Tania leaves out the
Solution 1
Tania writes 2019 just before leaving out 2020 (since 2020 is a multiple of 5).
Beginning at 1, 2020 is the 404
That is, the integers from 1 to 2020 contain 404 groups of 5 integers.
Each of these 404 groups contain one integer that is a multiple of 5, and so Tania leaves out 404 integers (including 2020) in the list of all integers from 1 to 2020.
If the
Solution 2
Tania writes 2019 just before leaving out 2020 (since 2020 is a multiple of 5).
Beginning at 1, 2020 is the 404
That is, the integers from 1 to 2020 contain 404 groups of 5 integers.
In Tania’s list of integers, she leaves out the integers that are multiples of 5, and so in every group of five integers, Tania lists four of these integers.
If the
Solution 1
We begin by determining which integers are in Tania’s list.
In each successive group of 5 consecutive integers beginning at 1, Tania lists 4 of the integers (since she leaves out each integer that is a multiple of 5).
That is, in each of these groups of 5 integers, Tania’s list contains
Consider all positive integers from 1 to
Of these
Tania’s list contains 200 integers, and so
That is, if Tania lists the positive integers from 1 to 250 and leaves out the integers that are multiples of 5, her list will contain
We are required to determine the sum,
We will proceed to determine this sum by first calculating the sum of all integers from 1 to 250, and then subtracting from that sum all integers in this list that are multiples of 5.
The sum of the integers from 1 to
The multiples of 5 in this list,
This sum is equal to
If Tania lists the positive integers, in order, leaving out the integers that are multiples of 5, the sum of the first 200 integers in her list is
Solution 2
As was shown in Solution 1, the sum of the first 200 integers in Tania’s list is the sum
The sum of the first and last integers in this list is
The sum of the second integer and the second last integer is
The sum of the third integer and the third last integer is
We continue in this way moving toward the middle of the list.
That is, we move one number to the right of the previous first number, and one number to the left of the previous second number.
Doing so, we recognize that
when the first number in the new pair is one more than the previous first number, then the number it is paired with is one less than the previous second number, and
when the first number in the new pair is two more than the previous first number (as is the case when a multiple of 5 is omitted), then the number it is paired with is two less than the previous second number.
That is, as we continue moving toward the middle of Tania’s list, each pair will continue to have a sum equal to 250.
Since there are 200 numbers in Tania’s list, there are 100 such pairs, each having a sum equal to 250.
Thus, if Tania lists the positive integers, in order, leaving out the integers that are multiples of 5, the sum of the first 200 integers in her list is
We begin by observing that when
Next, we will show that
In order for
This means
When
This can be checked on a calculator, but if we can understand why it is not a perfect square, we will be able to apply the reasoning again.
Suppose there is some integer
This means
However,
This means
However, neither
Therefore,
The key to the reasoning above is that
Thus, the same reasoning can be applied to conclude that
We now suppose
If
Therefore,
Thus,
Since
That is, for
We have now reduced the list of possible
Since
This means neither of
Since
We have now reduced the possibilities to
We will now show that
To do this, we write it as a product of prime numbers:
In particular, there are nine factors of
If
However, each factor of
Since
Therefore,
We have ruled out all other possibilities of
Building on the reasoning from part (a), we will use the observation that in any perfect square, there are an even number of factors of any given prime number.
For example,
The example given in the question has
Indeed, if
On the other hand, if a number factors into a product of primes where each prime occurs an even number of times, then the primes can be grouped to see that the number is a perfect square.
For example, the number with four factors of
In order for
By factoring
The primes 2,
From the discussion above, there must be an even number of each of these primes if the product is to be a perfect square.
Therefore, there must be another factor of each of the prime numbers
Furthermore, these primes must occur as factors of either
This means that one of
There are two variables and three primes, so one of
Since
Therefore, either
This means one of
Any positive multiple of
The variable which is not equal to
The only such multiples of
This forces
If
On the other hand, if
Therefore, the only Shonk sequence of the form
The longest Shonk sequence each of whose terms is an integer between
An example of such a sequence is
Each term after the first is greater than the previous term, so we need to verify that the product of all terms is a perfect square.
By splitting each term into a product of primes, we get
We will now show that no Shonk Sequence, each of whose terms is between
First, note that the only number between
Therefore, if
Similarly, if
This means the sequence must be made up of the numbers
There are
However, the product
Therefore, it cannot be a perfect square since a perfect square must have an even number of occurrences of any prime factor.
We need two related facts about perfect squares:
(F1) A positive integer
(F2) Suppose that integers
Suppose that
Since
Also, since
Therefore, if
Further, if
We can now answer the question by counting the number of pairs of positive integers
Note that
There are
We know that
Observe that
Thus, the possible values of
Thus, the number of pairs