Wednesday, April 5, 2023
(in North America and South America)
Thursday, April 6, 2023
(outside of North American and South America)
©2023 University of Waterloo
Jasmin’s total score was
Suppose that Sam had
Since Sam’s total score was 36 points, then
In total, Sam took
Since Tia’s total score was 37 points, then
Since
The value of
When
When
When
When
When
The possible ordered pairs
If
Since
Since 182 is not a multiple of 3, then it is not possible to have a
total score of 182 points.
The total area of the shaded regions can be determined by
subtracting the area of
The area of rectangle
The base length of
Thus, the total area of the shaded regions is
Solution 1
Since
The area of
Since the area of
The area of
The area of
Since the area of
Solution 2
Consider
The area of
The area of
Thus, the area of
The total area of the bottom two shaded regions (
The base length of
Since the area of
We can similarly show that the total area of the two top shaded
regions (
Since
Thus, the total area of the shaded regions is
Switching the second and fourth digits of 6238 gives the cousin
6832.
Since 6832 is not in the list, then it is the missing cousin.
Of the 16 six-digit integers in the given list, 15 are cousins of
the original integer.
If the original six-digit integer is
These 5 cousins are a result of switching the 1st digit,
Each of the remaining
Since the digits are distinct, then the 1st digit of the original
integer is the integer that occurs most frequently as the 1st digit in
the given list, which is 7.
A similar argument can be made for each of the other digits.
That is, the
Thus, the original integer has 2nd digit 2, 3rd digit 6, 4th digit 4,
5th digit 9, 6th digit 1, and so the original integer is 726 491.
We may check that if the original integer is 726 491, then the cousins
are indeed the remaining 15 integers in the given list.
The cousins of the three-digit integer
The difference between the two integers
Similarly, the difference between the two integers
Therefore,
Since the difference,
With
The three-digit integer
The three-digit integer
Thus,
(We may confirm that
The value of
Solution 1
The six cousins of the four-digit integer
The cousin
The cousin
The cousin
The cousin
The cousin
The cousin
The sum,
The ones digit of
If the ones digit of
Since
If
We continue to check the remaining possible pairs
52 527 | 30 747 | |
43 437 | 32 547 | |
34 347 | 34 347 | |
25 257 | 36 147 | |
16 167 | 37 947 | |
98 987 | 54 657 | |
89 897 | 56 457 |
The only pair of distinct, non-zero digits
Solution 2
The six cousins of the four-digit integer
Setting these two expressions equal to one another and simplifying, we
get
Since
Since
Substituting
Substituting
When
Since there are 9 ways to generate each of the three integers,
there are a total of
Next, we consider the different cases for which the product of the three
integers is a prime number.
The integers from 1 to 9 include the composite numbers 4, 6, 8, and
9.
If any one of the three integers generated is composite, then the
product of the three integers is composite.
Thus, the integers must be chosen from 1, 2, 3, 5, and 7.
Of these five integers, 2, 3, 5, and 7 are prime numbers.
If two or more of the three integers generated are prime numbers, then
the product of the three integers is composite.
Thus, at most one integer is a prime number.
If no integer generated is a prime number (all three are equal to 1),
then the product is 1, which is not prime.
Thus, Amarpreet’s product is a prime number exactly when one of the
integers is 2, 3, 5, or 7, and each of the other two integers is equal
to 1.
There are 4 ways to choose one of the prime numbers,
Therefore, the probability that the product is a prime number is
Solution 1
We begin by considering the different cases for which the product of
the four integers is divisible by 5, but not divisible by 7.
The only integer from 1 to 9 that is divisible by 5 is 5, and so the
product of the four integers is divisible by 5 exactly when at least one
of the four integers is a 5.
Similarly, the only integer from 1 to 9 that is divisible by 7 is 7, and
so the product of the four integers is not divisible by 7 exactly when a
7 is not among the four integers generated.
We proceed to count the number of ways to generate such arrangements of
four integers by considering the number of times that 5 appears in the
arrangement.
Case 1: there are four 5s
Since each of the four integers must be 5, there is 1 such possibility in this case.
Case 2: there are exactly three 5s
The three 5s can be arranged in 4 different ways:
There are 7 choices for the integer that is not a 5 since it can be any
of the nine integers, except 5 and 7.
Thus, there are
Case 3: there are exactly two 5s
The two 5s can be arranged in 6 different ways:
Once the two 5s have been placed, there are 7 choices for each of the
remaining two integers (since it is possible that these integers are
equal to one another).
Thus, there are
Case 4: there is exactly one 5
The 5 can be placed in 4 different ways.
Once the 5 has been placed, there are 7 choices for each of the
remaining three integers (since it is possible that these integers are
equal to one another).
Thus, there are
In total, there are
Since there are 9 ways to generate each of the four integers, there are
a total of
Therefore, the probability that Braxton’s product is divisible by 5, but
not divisible by 7 is
Solution 2
Let
Let
We are asked to find the probability of
Consider the Venn diagram shown in Figure 1.
The shaded region of this diagram represents
We wish to determine a more efficient method for determining
To do so, we proceed to use Venn diagrams to help express
Consider the next two Venn diagrams shown below.
In Figure 2, the shaded region represents
In Figure 3, the shaded region represents
Notice that if the region that is shaded in Figure 3 is removed
(unshaded) in Figure 2, then the resulting Venn diagram is equivalent to
that in Figure 1.
That is, mathematically,
(Without the use of the Venn diagrams, we may have similarly noticed
that since exactly one of
Thus, the probability that the product is divisible by 5, but not
divisible by 7, is equal to the probability that the product is not
divisible by 7, minus the probability that the product is both not
divisible by 5 and not divisible by 7.
The only integer from 1 to 9 that is divisible by 7 is 7, and so the
product of the four integers is not divisible by 7 exactly when a 7 is
not among the four integers generated.
The probability that the integer generated is not 7 is
The only integer from 1 to 9 that is divisible by 5 is 5, and so the
product of the four integers is not divisible by 5 exactly when a 5 is
not among the four integers generated.
The probability that the integer generated is not divisible by both 5
and 7 is thus
Therefore, the probability that the product is divisible by 5, but not
divisible by 7, is
Solution 3
Suppose that the four numbers generated are
When there are no restrictions, there are 9 choices for each of
If
If
This means that there are
Therefore, the probability that a random combination gives a product
that is not divisible by 7 but is divisible by 5 is
If any one of the 2023 integers generated is a 6, then the
product is divisible by 6.
Thus, we exclude 6 from the choice of possible integers.
Further, an integer is divisible by 6 exactly when it is divisible by
both 2 and 3.
With 6 excluded, the remaining integers that are divisible by 2 are 2, 4
and 8, and the remaining integers that are divisible by 3 are 3 and
9.
Therefore, the product is also divisible by 6 when at least one of the
integers generated is 2 or 4 or 8, and at least one of the integers
generated is 3 or 9.
To summarize, the product is not divisible by 6 exactly when
6 is not among the integers generated, and
there are integers from at most one of the two lists 2, 4, 8, and 3, 9.
Thus, there are exactly 3 cases which produce a product that is not divisible by 6, as follows.
Case | Is there a 6? | Is there a 2, 4 or 8? | Is there a 3 or 9? |
---|---|---|---|
1 | no | no | no |
2 | no | no | yes |
3 | no | yes | no |
For each of these 3 cases, we count the number of ways that the product generated is not divisible by 6.
Case 1: there is no 6, 2, 4, 8, 3, or 9
In this case, there are 3 integers that may be chosen (1, 5 and 7),
and so there are
Case 2: there is no 6, 2, 4, or 8
In this case, there are 5 integers that may be chosen (1, 3, 5, 7,
and 9), and so there are
However, this count includes the cases in which only 1, 5 and 7 are
chosen (3 and 9 are not), which were counted in Case 1.
Thus, the number of ways to generate the products in Case 2, different
from those in Case 1, is
Case 3: there is no 6, 3, or 9
In this case, there are 6 integers that may be chosen (1, 2, 4, 5, 7,
and 8), and there are
However, this count includes the cases in which only 1, 5 and 7 are
chosen (2, 4 and 8 are not), which were counted in Case 1.
Thus, the number of ways to generate the products in Case 3, different
from those in Case 1, is
Thus, the total number of ways to generate all products that are not
divisible by 6 is
Thus, the probability,
To determine the ones digit of the integer equal to
The ones digit of
The ones digit of
Consider the ones digit of successive integer powers of 3 beginning at
Since
Since
Finally, the ones digit of the integer equal to