Wednesday, May 12, 2021
(in North America and South America)
Thursday, May 13, 2021
(outside of North American and South America)
©2021 University of Waterloo
Arranging the five numbers from largest to smallest, we get
The middle number is 100.
Answer: (D)
Each side of the square has length 5 cm.
The perimeter of the square is
Answer: (A)
The right side of the equation is
The equation is true when the left side is also equal to 30.
Since
Answer: (E)
Reading from the graph, Dan spent 6 hours on homework, Joe spent 3 hours, Bob spent 5 hours, Susie spent 4 hours, and Grace spent 1 hour.
Adding their times together, Bob and Grace spent the same amount of time on homework as Dan.
Answer: (C)
Each of the five fractions is positive and so the smallest of these fractions is the fraction that is closest to 0.
Since each fraction has a numerator equal to 1, the smallest of these fractions is the one with the largest denominator.
Of those given, the fraction that is closest to 0 is thus
Answer: (E)
If the bag contained a total of 6 candies and exactly 5 of these candies were red, then the probability of Judith choosing a red candy from the bag would be
Therefore, the total number of candies in the bag could be 6.
Can you explain why each of the other four answers is not possible?
Answer: (D)
Each point that lies to the right of the
Each point that lies below the
Since
Answer: (B)
Begin by locating 2 km on the vertical (Distance) axis.
Next, locate the point on the line graph for which Andrew’s distance walked is 2 km, as shown.
The time in hours corresponding to this point is
Since
Answer: (C)
The five numbers
Thus, the 5
Since 220 is a multiple of 5, the 220
Answer: (A)
We begin by labelling additional points, as shown.
Beginning at
Assume the ant begins by travelling right, to
From
Thus, from
From
Therefore, there are two paths in which the ant begins by travelling right:
Assume the ant begins by travelling down, to
From
From
Therefore, there are two paths in which the ant begins by travelling down:
There are 4 different paths from
Answer: (C)
Solution 1
Writing the numbers that appear in the list, we get
Solution 2
Laila begins her list at 4 and each new number is 7 more than the previous number.
Therefore, each of the numbers in her list will be 4 more than a multiple of 7.
Since 42 is a multiple of 7 (
Answer: (B)
If a letter is folded along its vertical line of symmetry, both halves of the letter would match exactly.
Three of the given letters, H, O and X, have a vertical line of symmetry.
Answer: (C)
Since
Vertically opposite angles are equal in measure, and so
In
Thus,
Answer: (E)
Given three consecutive integers, the smallest integer is one less than the middle integer and the largest integer is one more than the middle integer.
For example, 10, 11 and 12 are three consecutive integers, and 10 is one less than the middle integer 11, and 12 is one more than 11.
So, the sum of the smallest and largest integers is twice the middle integer.
(In the example,
Then, the sum of three consecutive integers is equal to 3 times the middle integer and so the sum of three consecutive integers is a multiple of 3.
Of the answers given, the only number that is a multiple of 3 is 21.
Alternately, we could use trial and error to solve this problem to find that
Answer: (D)
There is no integer greater than 13931 and less than 14000 that is a palindrome. (You should consider why this is true before reading on.)
Let the next palindrome greater than 13931 be
We proceed under the assumption that
A 5-digit palindrome is a number of the form
Since
Since the smallest possible value of
Thus
Letting the hundreds digit,
Answer: (D)
The positive factors of 14 are
The positive factors of 21 are
The positive factors of 28 are
The positive factors of 35 are
The positive factors of 42 are
There are 3 numbers in the list (14, 21 and 35) that have exactly 4 positive factors.
Answer: (C)
The percentage discount of the third price off the original price does not depend on the original price of the shirt.
That is, we may choose an original price for the shirt and calculate the combined percentage discount.
Since the discounts are given as percentages, letting the original price of the shirt be $100 might make the calculations simpler.
If the original price of the shirt is $100 and this price is reduced by 50%, the discounted price is half of $100 or $50.
A further 40% reduction on $50 is equal to a
After both price reductions, the $100 shirt is priced at
The original price of the shirt was $100, the final discounted price is $70 less, and so the discount of the third price off the original price is
Answer: (C)
The perimeter of
Since
The perimeter of
The perimeter of
Since
Answer: (B)
We begin by calling the missing digits
Each of the digits
The top 2-digit number
Thus,
If
Thus, there are 4 possible positive results in this case.
If
In this case, the results of subtracting the bottom number from the top are 6, 7, 8, 10, 11, 12, 13, and 14 respectively. Thus, there are 8 possible positive results when
If
Thus, there are 8 possible positive results when
Similarly, when
In total, the number of possible results that are positive is
Answer: (A)
The table below shows the possible sums when two standard dice are rolled.
Each sum in bold is equal to a prime number.
Number on the First Die | |||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
Number on the Second Die | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | 3 | 4 | 5 | 6 | 7 | 8 | |
3 | 4 | 5 | 6 | 7 | 8 | 9 | |
4 | 5 | 6 | 7 | 8 | 9 | 10 | |
5 | 6 | 7 | 8 | 9 | 10 | 11 | |
6 | 7 | 8 | 9 | 10 | 11 | 12 |
Looking at the table above, the total number of possible outcomes is
The total number of outcomes for which the sum is a prime number is 15.
The probability that the sum of the numbers on the top faces is a prime number is
Answer: (A)
We begin by considering cases in which 1 is subtracted from numbers that start with one and are followed by smaller numbers of zeros.
In the examples above, each result consists of 9s only and the number of 9s is equal to the number of zeros in the original number. Can you explain why this pattern continues as we increase the number of zeros?
Since each of the digits in the result is a 9 and the sum of these digits is 252, then the number of 9s in the result is equal to
The number of zeros in the original number equals the number of 9s in the result, which is 28.
Answer: (B)
The perimeter of Figure 1 consists of 4 rectangle side lengths of 10 cm (each of which is horizontal) and 4 rectangle side lengths of 5 cm (each of which is vertical).
Thus, the perimeter of Figure 1 is
The perimeter of Figure 2 consists of 4 rectangle side lengths of 10 cm (each of which is horizontal) and 6 rectangle side lengths of 5 cm (each of which is vertical).
Thus, the perimeter of Figure 2 is
The perimeter of Figure 3 consists of 4 rectangle side lengths of 10 cm (each of which is horizontal) and 8 rectangle side lengths of 5 cm (each of which is vertical).
Thus, the perimeter of Figure 3 is
Each figure after Figure 1 is formed by joining two rectangles to the bottom of the previous figure.
The bottom edge of a figure (consisting of two 10 cm side lengths) is replaced by two 10 cm lengths when the two new rectangles are adjoined.
That is, the addition of two rectangles does not change the number of 10 cm side lengths contributing to the perimeter of the new figure and so the number of 10 cm lengths remains constant at 4 for each figure.
The addition of two new rectangles does not replace any of the previous 5 cm (vertical) side lengths.
Thus, the addition of the two rectangles does add two 5 cm vertical segments to the previous perimeter, increasing the perimeter of the previous figure by
That is, the perimeter of Figure 1 is 60 cm, and the perimeter of each new figure is 10 cm greater than the previous figure.
We need to add 10 cm 65 times to get a total of 710 cm (that is,
Thus, Figure 66 has a perimeter of 710 cm, and so
Answer: (C)
To encode a letter, James multiplies its corresponding number by 3 and then subtracts 5, continuing this process a total of
To decode a number, the inverse operations must be performed in the opposite order.
The inverse operation of multiplication is division. The inverse operation of subtraction is addition.
Thus to decode a number, add 5 and then divide by 3, and continue this process a total of
For example, when
Each letter of James’ original message corresponds to a number from 1 to 26, inclusive.
To determine the value of
We show this work in the table below.
Encoded number | 367 | 205 | 853 | 1339 |
---|---|---|---|---|
16 | 10 | 34 | 52 | |
7 | 5 | 13 | 19 | |
4 | 6 | 8 |
From the table, the first value of
Further, we note that if
Therefore, the value of
(Although the question did not ask for the original message, the letters corresponding to
Answer: (C)
We begin by considering the prime factors (called the prime factorization) of each of the two numbers, 4 and 4620.
Since the pair
Further, each of
The lowest common multiple of
Further, if for example
That is, each of
Summarizing,
the prime factors of
the prime factors of
In the table below, we list the possible values for
To ensure that we don’t double count the pairs
There are 8 different pairs of positive whole numbers having a greatest common factor of 4 and a lowest common multiple of 4620.
Answer: (D)
Since
The net contains the numbers 100 and
Jonas builds the large cube in such a way that the sum of the numbers on the exterior faces is as large as possible.
Since
The
We call these three types: corner, edge and inside. In the portion of the large
A corner cube is shown in Figure 1. These are cubes that appear in one of the “corners” of the large cube and so there are 8 such corner cubes.
An edge cube is shown in Figure 2. These are cubes that appear along the edges but not in the corners of the large cube.
A cube has 12 edges and each edge of the large cube contains 10 edge cubes, and so there are
An inside cube is shown in Figure 3. These are the remaining cubes that contribute to the numbers on the exterior faces of the large cube.
A cube has 6 faces and each face of the large cube contains
Let
Each corner cube has 3 faces which contribute to
Thus, the 8 corner cubes contribute
Each edge cube has 2 faces which contribute to
For
Thus, the 120 edge cubes contribute
Finally, each inside cube has 1 face which contributes to
For
Thus, the 600 inside cubes contribute
In total, we get
Since we want
Because
Since we want
Because
This means that
There are
Answer: (C)
Since 1000 is 1 more than 999, then
Thus,
Answer: (C)
An equilateral triangle has 3 sides of equal length.
If the perimeter of an equilateral triangle is 15 m, then the length of each side is
Answer: (B)
Since
Therefore, the greatest multiple of 4 less than 100 is
Answer: (B)
Points which lie to the right of the
Points which lie below the
Point
Answer: (B)
Substituting
(A):
(B):
(C):
(D):
(E):
Of these,
Answer: (B)
At this rate, it would take 6 seconds to fill a 500 mL bottle.
A 250 mL bottle has half the volume of a 500 mL bottle and so it will take half as long or 3 seconds to fill.
Answer: (C)
If the tens digit of a two-digit number is even, then when the digits are reversed the new number will have a units digit that is even and therefore the number will be even.
If a two-digit number is even, then it is divisible by 2 and so it cannot be a prime number.
Since 29, 23 and 41 each have a tens digit that is even, we may eliminate these three as possible answers.
When the digits of 53 are reversed, the result is 35.
Since 35 is divisible by 5, it is not a prime number.
Finally, when the digits of 13 are reversed, the result is 31.
Since 31 has no positive divisors other than 1 and 31, it is a prime number.
Answer: (D)
When 3 red beans are added to the bag, the number of red beans in the bag is
When 3 black beans are added to the bag, the number of black beans in the bag is
The number of beans now in the bag is
If one bean is randomly chosen from the bag, the probability that the bean is red is
Answer: (B)
We begin by labelling additional points, as shown.
Beginning at
Assume the ant begins by travelling right, to
Therefore, there are two paths in which the ant begins by travelling right:
Assume the ant begins by travelling down, to
Thus, from
Therefore, there are two paths in which the ant begins by travelling down:
There are 4 different paths from
Answer: (C)
By assigning the largest digits to the largest place values, we form the largest possible four-digit number.
The largest four-digit number that can be formed by rearranging the digits of 2021 is 2210.
By assigning the smallest digits to the largest place values, we form the smallest possible four-digit number.
The smallest four-digit number (greater than 1000) that can be formed by rearranging the digits of 2021 is 1022.
Thus the largest possible difference between two such four-digit numbers is
Answer: (A)
Solution 1
Since
and so the measure of
Solution 2
Answer: (C)
Given three consecutive integers, the smallest integer is one less than the middle integer and the largest integer is one more than the middle integer.
So, the sum of the smallest and largest integers is twice the middle integer.
Then, the sum of three consecutive integers is equal to 3 times the middle integer and so the sum of three consecutive integers is a multiple of 3.
Of the answers given, the only number that is a multiple of 3 is 21.
Alternately, we could use trial and error to solve this problem to find that
Answer: (D)
Reading from the bar graph, there are 8 yellow shirts, 4 red shirts, 2 blue shirts, and 2 green shirts. In total, the number of shirts is
Thus, 8 yellow shirts represents
The only circle graph showing that approximately half of the shirts are yellow is (E) and thus it probably best represents the information in the bar graph.
We may confirm that this circle graph also shows that approximately
Answer: (E)
Let the unknown whole number be
Since 16 is a factor of
That is, the positive factors of
Since 16 is a factor of
The smallest whole number multiple of 16 is 16.
However, if
The next smallest whole number multiple of 16 is 32.
If
Answer: (B)
Solution 1
The sum of the three interior angles in a triangle is
A triangle’s three interior angles are in the ratio
Thus, the smallest angle in the triangle measures
The measure of the next largest angle is 4 times the measure of the smallest angle or
The measure of the largest angle is 7 times the measure of the smallest angle or
The measures of the interior angles are
Solution 2
The sum of the three interior angles in a triangle is
Working backward from the possible answers, we may eliminate (B) and (C) since the sum of the three given angles is not
The measures of the smallest and largest angles in the triangle are in the ratio
Since
Since
The remaining possibility is (D) and we may confirm that
Answer: (D)
The seven numbers
Thus, the 7
The 14
Since 70 is a multiple of 7, the 70
The sum of the 18
Answer: (E)
Solution 1
Gaussville’s soccer team won 40% of their first 40 games.
Thus they won
After winning the next
At this point, they had won 50% or
This means that the number of games won,
For which of the possible answers is
Substituting each of the five possible answers, we get that
Solution 2
Gaussville’s soccer team won 40% of their first 40 games.
Thus they had
At this point, the team went on a winning streak which means they did not accumulate any additional non-wins.
Thus, their 24 non-wins represent 50% of the final total, and so the final number of wins is 24.
Therefore, Gaussville’s soccer team won
Answer: (D)
The fraction of the area of the larger circle that is not shaded does not depend on the actual radius of either circle, and so we begin by letting the radius of the smaller circle be 1 and thus the radius of the larger circle is 3.
In this case, the area of the smaller circle is
The area of the larger circle is
The area of the larger circle that is not shaded is
Therefore, the fraction of the area of the larger circle that is not shaded is
(Alternately, we could note that the fraction of the area of the larger circle that is shaded is
Answer: (A)
We proceed to work backward from the final sum, 440, ‘undoing’ each of the three operations to determine the sum of their two numbers before any operations were performed.
The final operation performed by each of Asima and Nile was to multiply their number by 4.
Multiplying each of their numbers by 4 increases the sum of the two numbers by a factor of 4.
That is, the final sum of their two numbers was 440, and so the sum of their two numbers immediately before the last operation was performed was
The second operation performed by each of Asima and Nile was to subtract 10 from their number.
Subtracting 10 from each of their numbers decreases the sum of the two numbers by 20.
That is, the sum of their two numbers immediately following the second operation was 110, and so the sum of their two numbers immediately before the second operation was performed was
Finally, the first operation performed by each of Asima and Nile was to double their number.
Doubling each of their numbers increases the sum of the two numbers by a factor of 2.
That is, the sum of their two numbers immediately following the first operation was 130, and so the sum of their two numbers before the first operation was performed was
Each of their original integers is greater than 0 and the two integers have a sum of 65.
Therefore, Asima’s original integer could be any integer from 1 to 64, inclusive.
Thus, there are 64 possibilities for Asima’s original integer. Answer: (A)
Solution 1
The table below shows the possible differences between the number on Ruby’s roll and the number on Sam’s roll.
Number on Ruby's Roll | |||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
Number on Sam's Roll | 1 | ||||||
2 | |||||||
3 | |||||||
4 | |||||||
5 | |||||||
6 |
Looking at the table above, the total number of possible outcomes is
The total number of outcomes for which the difference is negative is 15.
The probability that the result from subtracting the number on Sam’s roll from the number on Ruby’s roll is negative is
Solution 2
Ruby and Sam each have 6 possible outcomes when they roll the dice, and so the total number of possible outcomes is
Of these 36 possible outcomes, there are 6 outcomes in which Sam and Ruby each roll the same number and thus the difference between the numbers rolled is 0.
For the remaining
That is, one half of these 30, or 15 possible outcomes have a result that is negative and 15 have a result that is positive.
Therefore, the probability that the result from subtracting the number on Sam’s roll from the number on Ruby’s roll is negative is
Answer: (B)
If
For example,
2021 0s
The result of adding 1 to the positive integer consisting of only
For example,
2021 9s 2021 0s
Let
Since
2021 9s 2021 9s 2021 9s 2017 9s
The sum of the digits of the integer equal to
Answer: (E)
We begin by listing the prime numbers up to and including 31.
We choose to end the list at 31 since
This list of prime numbers is
There are 3 positive integers less than 900 that can be written as a product of three consecutive prime numbers. These are
Further, we note that the next smallest positive integer that can be written as a product of three consecutive prime numbers is
Exactly one positive integer less than 900 can be written as a product of four consecutive prime numbers. This number is
There are no positive integers that can be written as a product of five or more consecutive prime numbers since
In total, the number of positive integers less than 900 that can be written as a product of two or more consecutive prime numbers is
Answer: (A)
We begin by labelling some additional points as shown.
With the leash extended the full 4 m, the dog can reach points
The doghouse
Therefore, the shaded figure is
There is additional area outside the doghouse in which the dog can play, as shaded in the diagram.
Since
Similarly,
Since
The area outside of the doghouse in which the dog can play is
Answer: (A)
Let the sum of the numbers on the exterior faces of the
To determine the smallest value of
The
We call these three types: corner, edge and inside.
In the portion of the large
A corner cube is shown in Figure 1. These are cubes that appear in one of the “corners” of the large cube and so there are 8 such corner cubes.
An edge cube is shown in Figure 2. These are cubes that appear along the edges but not in the corners of the large cube.
A cube has 12 edges and each edge of the large cube contains
An inside cube is shown in Figure 3. These are the remaining cubes that contribute to the numbers on the exterior faces of the large cube.
A cube has 6 faces and each face of the large cube contains
Each corner cube has 3 faces which contribute to
For
We may determine from the given net that the three faces meeting at a vertex of the
The sums of these three faces are
To make
Therefore, the 8 corner cubes contribute
Each edge cube has 2 faces which contribute to
We may determine from the given net that two faces which share an edge of the
The sums of these two adjacent faces are
To make
Therefore, the
Finally, each inside cube has 1 face which contributes to
Thus, the
In total, we get
We want the smallest value of
Using trial and error with the given answers, we get the following:
When
When
When
When
The smallest value of
Answer: (D)
We begin by drawing a well-labelled diagram, as shown.
Since
The value of
The value of
Thus, the value of
Since the areas of rectangles
The area of
Similarly, the area of
For what values of
If
Similarly,
For each of the 7 possible values of
As an example, consider
In this case,
Having considered the cases in which both
Specifically, we will assume that
Are there non-integer values of
When
Let
Since
Next, we consider each of these possible values for
When
Integer values of both
Next, we consider the cases for which
Since
Further,
When
Recall that we are considering cases in which exactly one of
The area of rectangle
When
Since
For each of the 8 choices for
As an example, consider
Recall from earlier that
Thus, each of the possible values of
That is,
To summarize to this point, if both
If
If
Next, we consider the cases for which
Since
Further,
When
Recall that we are considering cases in which exactly one of
Since
Since
For each of the 16 choices for
As an example, consider
In this case,
When
To complete the cases for which exactly one of
Since
However,
Finally, we consider cases in which both
As previously determined, if
If
However, if
Since it is not possible for 4, 8 or 16 to be a factor of an odd number, then
Thus, there are no cases for which both
Therefore, the value of
Answer: (D)