Wednesday, May 13, 2020
(in North America and South America)
Thursday, May 14, 2020
(outside of North American and South America)
©2020 University of Waterloo
One pen costs $2. The cost of 10 pens is
Answer: (E)
Beginning at the origin
Answer: (E)
Since 99 is close to 100 and 9 is close to 10, the value of
Of the given answers, this means that the integer that is closest to
Multiplying, we get that the value of
The increase in temperature equals the warmer afternoon temperature minus the colder morning temperature.
The difference between 5 and
Answer: (A)
In April, Alexis averaged
Answer: (B)
Solution 1
One complete rotation equals
Since
Therefore, the remaining
Solution 2
As in Solution 1,
Since the 80 students who chose juice represent
Answer: (C)
Since the third and fourth numbers in the list are consecutive and add to 11, then the third number in the list is 5 and the fourth is 6. The fifth number in the list is 7, and so the sixth number in the list is 8.
Answer: (D)
Solution 1
Between 0 and 1.0, the number line is divided by tick marks (at
Thus, the distance between adjacent tick marks is
Since
Since
Solution 2
The number
The number
The tick marks are equally spaced along the number line, and so the value of
Answer: (B)
The triangle is isosceles, and so the missing side length is also 12 cm.
The perimeter of the triangle is
The perimeter of the rectangle is given by
The perimeter of the triangle is equal to the perimeter of the rectangle, and so
Answer: (C)
In the table below, we list the divisors of each of the given answers (other than the number itself) and determine their sum.
Given Answers | Divisors | Sum of the Divisors |
---|---|---|
8 | ||
10 | ||
14 | ||
18 | ||
22 |
The sum of the divisors of each of
Each of these four answers is not an abundant number. (These numbers are called “deficient”.)
The sum of the divisors of 18 is 21, which is greater than 18. Thus 18 is an abundant number.
Answer: (D)
Each of 7 boxes contains exactly 10 cookies, and so the total number of cookies is
Answer: (A)
Abdul is 9 years older than Susie, and Binh is 2 years older than Susie, and so Abdul is
Answer: (E)
The
Therefore, the distance between
Similarly, the
Therefore, the distance between
Answer: (C)
Before Jack eats any jelly beans, the box contains
After Jack eats 2 jelly beans, there are
One of the jelly beans that Jack ate was green, and the other was blue.
Thus after eating the 2 jelly beans, there are still 15 red jelly beans in the box.
If each of the remaining jelly beans is equally likely to be chosen, the probability that Jack chooses a red jelly bean next is
Answer: (C)
Solution 1
There are 60 minutes in 1 hour, and so there are
If Emil’s race time was 54 minutes, then Olivia’s race time was 4 minutes more, or 58 minutes.
In this case, their race times total
Solution 2
As in Solution 1, the total of their race times is 112 minutes.
If Emil’s race time was 4 minutes more, then his race time would be equal to Olivia’s, and the total of their race times would be
If their race times are equal and total 116 minutes, then they each finished the race in
Answer: (C)
For
To see why this is, consider that the reflection in
Further, the reflection in
Thus, the reflection in
More generally, the reflection in
Answer: (A)
If Rosie deposits $30 each month for
Answer: (E)
Isosceles triangles have two equal angles, and so the possibilities for these two triangles are:
1) The two equal angles are each equal to
2) The two equal angles are each not equal to
(We note that a triangle can not have three angles measuring
If the two equal angles are each equal to
If the two equal angles are each not equal to
We note that in the first triangle, the measure of each of the two remaining angles (
In the first triangle, the sum of the two equal angles is
In the second triangle, the sum of the two equal angles is
Answer: (B)
We begin by recognizing that there are 6 different symbols, and so each face of the cube contains a different symbol.
From left to right, let us number the views of the cube 1, 2 and 3.
Views 1 and 2 each show a face containing the symbol .
What symbol is on the face opposite to the face containing ?
In view 1, and
are on faces adjacent to the face containing
, and so neither of these can be the symbol that is on the face opposite
.
In view 2, and
are on faces adjacent to the face containing
, and so neither of these can be the symbol that is on the face opposite
.
There is only one symbol remaining, and so must be the symbol that is on the face opposite
, and vice versa.
A net of the cube is shown below.
Answer: (C)
On each of her four tosses of the coin, Jane will either move up one dot or she will move right one dot.
Since Jane has two possible moves on each of her four tosses of the coin, she has a total of
If we denote a move up one dot by
The probability of tossing a head (and thus moving up one dot) is equal to the probability of tossing a tail (and thus moving right one dot).
That is, it is equally probable that Jane will take any one of these 16 paths.
Therefore, the probability that Jane will finish at dot
How many of the 16 paths end at dot
Beginning at
There are 6 such paths:
(We note that each of the other 10 paths will end at one of the other 4 dots,
Answer: (B)
Each four-digit number must be greater than 2000, and so the smallest two-digit number that may be repeated is 20 (to give 2020).
Each four-digit number must be less than 10 000, and so the largest two-digit number that may be repeated is 99 (to give 9999).
Each two-digit number between 20 and 99 may be repeated to give a four-digit number between 2000 and 10 000.
In total, there are
Answer: (A)
Celyna spent $5.00 on candy A and $7.00 on candy B, or $12.00 in total.
The average price of all the candy that she purchased was $1.50 per 100 grams.
This means that if Celyna bought 100 grams of candy, she would have spent $1.50.
If she bought 200 grams of candy, she would have spent
How many grams of candy would Celyna need to buy to spend $12.00?
Since
Answer: (C)
If the first positive integer in the list is
Thus, we are asked to find the number of pairs of positive integers
What is the largest possible value for
If
If
However in this case, we get that
If
Thus, the largest possible value for
What is the smallest value for
If
If
However in this case, we get that
If
However, if the first integer in the list is 26, then the second integer can not equal 24 since the list is increasing.
Smaller values of
From the values of
So when
However,
Conversely, when
Here are the 8 lists:
Answer: (E)
We begin by determining the area of
The base
We wish to determine which of the 41 points is a possible location for
The base
Since the distance between two parallel lines remains constant, any point
The line labelled
Next, we consider the possible locations for
Since
However, if
For this reason,
If we consider the base of
Any point
(This is the same property that we saw previously for
How do we create a line that passes through
Beginning at
Begin at
Can you explain why the line segment
Any point that lies on
There are 2 points that lie on
(We note that we may move right 5 and up 10 by moving in ‘steps’ of right 1 and up 2 to arrive at each of these 2 points.)
These 2 points and one possibility for
Finally, we consider the possible locations for
As a result of symmetry, this case is identical to the previous case.
Thus, there are 2 possible locations for
Answer: (E)
Bus A takes 12 minutes to complete one round trip that begins and ends at
Since
That is, Bus A first arrives at
Bus A | 1:03 | 1:09 | 1:15 | 1:21 | 1:27 | 1:33 | 1:39 | 1:45 | 1:51 | 1:57 | 2:03 |
---|
Notice that Bus A arrives at
This makes sense since Bus A returns to
This tells us that Bus A will continue to arrive at the same number of minutes past each hour, or 2:03, 2:09, 2:15,
Bus B takes 20 minutes to complete one round trip that begins and ends at
Since
That is, Bus B first arrives at
Bus B | 1:05 | 1:15 | 1:25 | 1:35 | 1:45 | 1:55 | 2:05 |
---|
Notice that Bus B arrives at
This makes sense since Bus B returns to
This tells us that Bus B will continue to arrive at the same number of minutes past each hour, or 2:05, 2:15, 2:25,
From the two tables above, we see that Bus A and Bus B both arrive at
Thus between 5:00 p.m. and 10:00 p.m., these two buses will meet
Bus C takes 28 minutes to complete one round trip that begins and ends at
Since
That is, Bus C first arrives at
Unlike Bus A and Bus B, Bus C will not arrive at
What is the first time after 5:00 p.m. that Bus C arrives at
Since 238 is a multiple of 14 (
Since 238 minutes is 2 minutes less than 4 hours (
Bus B also arrives at
Are there other times after 5:05 p.m. (and before 10:00 p.m.) that Bus B and Bus C arrive at
Bus B arrives at
Since the lowest common multiple of 10 and 14 is 70, then Bus B and Bus C will each arrive at
Next we determine if there are times when Bus A and Bus C arrive at
Bus C arrives at
Bus A also arrives at
Are there other times after 5:33 p.m. (and before 10:00 p.m.) that Bus A and Bus C arrive at
Bus A arrives at
Since the lowest common multiple of 6 and 14 is 42, then Bus A and Bus C will each arrive at
The times when each pair of buses meet at
Bus A and Bus B: 15 and 45 minutes past each hour
Bus B and Bus C: 5:05 p.m., 6:15 p.m., 7:25 p.m., 8:35 p.m., 9:45 p.m.
Bus A and Bus C: 5:33 p.m., 6:15 p.m., 6:57 p.m., 7:39 p.m., 8:21 p.m., 9:03 p.m., 9:45 p.m.
Finally, we determine the number of different times that two or more buses arrive at
Bus A and Bus B arrive at
Bus B and Bus C arrive at
Bus A and Bus C arrive at
Answer: (A)
Including 1 with the four given numbers and ordering the list, we get
Answer: (D)
If the total cost of 4 one-litre cartons of milk is $4.88, then the cost of 1 one-litre carton of milk is
Answer: (E)
Of the answers given,
Answer: (E)
Since
Answer: (E)
The length of the base is the distance from the origin to the point
The height is the distance from the origin to the point
Thus, the area of the triangle is
Answer: (A)
The whole numbers between 2 and 20 whose square root is a whole number are:
We may note that perfect squares are equal to the square of an integer.
That is, the smallest five positive perfect squares are
Answer: (D)
Solution 1
For each of the 4 different notebooks that Yvonn may choose, there are 5 different pens that may be chosen. Thus, there are
Solution 2
We begin by naming the 4 notebooks
If Yvonn chooses notebook
If Yvonn chooses notebook
If Yvonn chooses notebook
Similarly, Yvonn has 5 pen choices for each choice of notebook.
These remaining possible combinations are:
Answer: (C)
We begin by recognizing that a right angle is marked in the pie chart.
Since
Answer: (B)
There are 8 letters in the bag and 2 of these letters are
Answer: (A)
Balil’s result,
Cali’s result,
Thus the difference between Balil’s result and Cali’s result,
Answer: (E)
The bus stops at the library at 1:00 p.m., 1:20 p.m., and 1:40 p.m..
The bus stops at the library at 2:00 p.m., 2:20 p.m., and 2:40 p.m..
Similarly, the bus stops at the library at
Finally, the bus stops at 6:00 p.m..
In total, the bus stops at the library
Answer: (A)
Solution 1
Beginning with the units (ones) column, the units digit of the sum
Thus,
If
In this case, the units digit of the sum
This is not possible since
Thus,
In the tens column the units digit of the sum
If
This is not possible since the sum of the hundreds column is 10 and
Thus,
The sum of the hundreds column is 10, and so
Solution 2
The sum of
Since the sum of the hundreds column is 10, the ‘carry’ from the tens column must be 1.
Considering the tens and units (ones) columns together, we see that the result of
Answer: (E)
Solution 1
There are 60 minutes in 1 hour, and so there are
If Emil’s race time was 54 minutes, then Olivia’s race time was 4 minutes more, or 58 minutes.
In this case, their race times total
Solution 2
As in Solution 1, the total of their race times is 112 minutes.
If Emil’s race time was 4 minutes more, then his race time would be equal to Olivia’s, and the total of their race times would be
Answer: (C)
In
Answer: (D)
If Francesca first chooses
If she first chooses
If she first chooses
If she first chooses
If she first chooses
If she first chooses
If Francesca first chooses
Answer: (B)
Since
Opposite angles are equal in measure, and so
Answer: (C)
In a list of five numbers ordered from smallest to largest, the median is equal to the third (middle) number in the list.
Since
each of
each of
at least one of
If each of
(In this case, the ordered list could be
If each of
(In this case, the ordered list could be
If at least one of
When the list is ordered, the two 12s will either be the 2nd and 3rd numbers in the list, or the 3rd and 4th numbers in the list depending on whether the unknown number is less than or equal to 12 or greater than or equal to 12.
In either case, the median is 12. Thus, there are three different possible medians for Mark’s five point totals.
Answer: (C)
We begin by recognizing that there are 6 different symbols, and so each face of the cube contains a different symbol.
From left to right, let us number the views of the cube 1, 2 and 3.
Views 1 and 2 each show a face containing the symbol .
What symbol is on the face opposite to the face containing ?
In view 1, and
are on faces adjacent to the face containing
, and so neither of these can be the symbol that is on the face opposite
.
In view 2, and
are on faces adjacent to the face containing
, and so neither of these can be the symbol that is on the face opposite
.
There is only one symbol remaining, and so must be the symbol that is on the face opposite
, and vice versa.
A net of the cube is shown below.
Answer: (C)
20% of 100 is 20, and so 20% of 200 is 40. Thus
40 is
(We may check that 80% of 50 is indeed
Answer: (D)
We begin by expressing
Converting
Since
The numerators of
That is,
Since
Answer: (B)
Originally, the ratio of green balls to yellow balls in the bag was
This means that for every 3 green balls in the bag, there were 7 yellow balls.
Equivalently, if there were
After 9 balls of each colour are removed, the number of green balls in the bag is
At this point, the ratio of green balls to yellow balls is
Multiplying the number of green balls by 3, we get
Solving the equation
Originally, there were
Answer: (B)
A number is divisible by 6 if it is divisible by both 2 and 3.
To be divisible by 2, the three-digit number that is formed must be even and so the ones digit must be 0 or 2.
To be divisible by 3, the sum of the digits of the number must be a multiple of 3.
Consider the possible tens and hundreds digits when the ones digit is 0.
In this case, the sum of the tens and hundreds digits must be a multiple of 3 (since the ones digit does not add anything to the sum of the digits).
We determine the possible sums of the tens and hundreds digits in the table below. The sums which are a multiple of 3 are circled.
The Tens Digit | |||||
---|---|---|---|---|---|
5 | 6 | 7 | 8 | ||
The Hundreds Digit | 1 | ⑥ | 7 | 8 | ⑨ |
2 | 7 | 8 | ⑨ | 10 | |
3 | 8 | ⑨ | 10 | 11 | |
4 | ⑨ | 10 | 11 | ⑫ |
When the ones digit is 0, the possible three-digit numbers are:
Consider the possible tens and hundreds digits when the ones digit is 2.
In this case, the sum of the tens and hundreds digits must be 2 less than a multiple of 3 (since the ones digit adds 2 to the sum of the digits).
When the ones digit is 2, the possible three-digit numbers are:
Answer: (A)
We begin by joining the centre of the rectangle,
Since
Similarly,
Thus, quadrilateral
Similarly, we can show that quadrilateral
Answer: (D)
Bus A takes 12 minutes to complete one round trip that begins and ends at
Since
That is, Bus A first arrives at
Bus A | 1:03 | 1:09 | 1:15 | 1:21 | 1:27 | 1:33 | 1:39 | 1:45 | 1:51 | 1:57 | 2:03 |
---|
Notice that Bus A arrives at
This makes sense since Bus A returns to
This tells us that Bus A will continue to arrive at the same number of minutes past each hour, or 2:03, 2:09, 2:15,
Bus B takes 20 minutes to complete one round trip that begins and ends at
Since
That is, Bus B first arrives at
Bus B | 1:05 | 1:15 | 1:25 | 1:35 | 1:45 | 1:55 | 2:05 |
---|
Notice that Bus B arrives at
This makes sense since Bus B returns to
This tells us that Bus B will continue to arrive at the same number of minutes past each hour, or 2:05, 2:15, 2:25,
From the two tables above, we see that Bus A and Bus B both arrive at
Thus between 5:00 p.m. and 10:00 p.m., these two buses will meet
Bus C takes 28 minutes to complete one round trip that begins and ends at
Since
That is, Bus C first arrives at
Unlike Bus A and Bus B, Bus C will not arrive at
What is the first time after 5:00 p.m. that Bus C arrives at
Since 238 is a multiple of 14 (
Since 238 minutes is 2 minutes less than 4 hours (
Bus B also arrives at
Are there other times after 5:05 p.m. (and before 10:00 p.m.) that Bus B and Bus C arrive at
Bus B arrives at
Since the lowest common multiple of 10 and 14 is 70, then Bus B and Bus C will each arrive at
Next we determine if there are times when Bus A and Bus C arrive at
Bus C arrives at
Bus A also arrives at
Are there other times after 5:33 p.m. (and before 10:00 p.m.) that Bus A and Bus C arrive at
Bus A arrives at
Since the lowest common multiple of 6 and 14 is 42, then Bus A and Bus C will each arrive at
The times when each pair of buses meet at
Bus A and Bus B: 15 and 45 minutes past each hour
Bus B and Bus C: 5:05 p.m., 6:15 p.m., 7:25 p.m., 8:35 p.m., 9:45 p.m.
Bus A and Bus C: 5:33 p.m., 6:15 p.m., 6:57 p.m., 7:39 p.m., 8:21 p.m., 9:03 p.m., 9:45 p.m.
Finally, we determine the number of different times that two or more buses arrive at
Bus A and Bus B arrive at
Bus B and Bus C arrive at
Bus A and Bus C arrive at
Answer: (A)
The property of an integer being either even or odd is called its parity.
If two integers are both even or they are both odd, then we say that the two integers have the same parity.
If one integer is even and a second integer is odd, then we say that the two integers have different parity.
The result of adding two integers that have the same parity is an even integer.
The result of adding two integers that have different parity is an odd integer.
The parity of each term of an FT sequence (after the second term) is determined by the parity of the first two terms in the sequence.
For example, if each of the first two terms of an FT sequence is odd, then the third term is even (since odd plus odd is even), the fourth term is odd (since odd plus even is odd), the fifth term is odd (since even plus odd is odd), and so on.
There are 4 possibilities for the parities of the first two terms of an FT sequence.
The sequence could begin odd, odd, or even, even, or odd, even, or even, odd. In the table below, we write the parity of the first few terms of the FT sequences that begin in each of the 4 possible ways.
Term Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Parity #1 | odd | odd | even | odd | odd | even | odd | odd | even | odd |
Parity #2 | even | even | even | even | even | even | even | even | even | even |
Parity #3 | odd | even | odd | odd | even | odd | odd | even | odd | odd |
Parity #4 | even | odd | odd | even | odd | odd | even | odd | odd | even |
The FT sequence beginning odd, odd (Parity #1) continues to repeat odd, odd, even.
Since the parity of each term is dependent on the parity of the two terms preceding it, this odd, odd, even pattern will continue throughout the entire sequence.
That is, in each successive group of three terms beginning at the first term, one out of three terms will be even and two out of three terms will be odd.
The odd, odd, even pattern ends at term numbers that are multiples of 3 (the even-valued terms are terms 3, 6, 9, 12, and so on).
Since 2019 is a multiple of 3 (
The 2020
This is exactly the required condition for the FT sequences that we are interested in.
How many FT sequences begin with two odd-valued terms, each of which is a positive integer less than
There are
Since
Thus, the list of integers from 1 to
The first term in the sequence is odd-valued and so there are
Similarly, the second term in the sequence is also odd-valued and so there are also
Do any of the other 3 types of FT sequences satisfy the required condition that there are more than twice as many odd-valued terms as there are even-valued terms?
Clearly the FT sequence beginning even, even (Parity #2) does not satisfy the required condition since every term in the sequence is even-valued.
The FT sequence beginning odd, even (Parity #3) continues to repeat odd, even, odd.
That is, in each successive group of three terms beginning at the first term, one out of three terms will be even and two out of three terms will be odd.
As we saw previously, 2019 is a multiple of 3 and so
Thus there are twice as many odd-valued terms as there are even-valued terms in the first 2019 terms.
The 2020
How many FT sequences begin with an odd-valued first term and an even-valued second term, each being a positive integer less than
As we showed previously, the list of integers from 1 to
The first term in the sequence is odd-valued and so there are
The second term in the sequence is even-valued and so there are
Finally, we consider the FT sequences that begin with an even-valued term followed by an odd-valued term (Parity #4).
Again, there are exactly twice as many odd-valued terms as there are even-valued terms in the first 2019 terms (since the pattern repeats even, odd, odd).
However in this case, the 2020
Thus, there are
Since there are 2415 such FT sequences, we may solve
Evaluating
When
When
Answer: (D)