Thursday, February 23, 2021
(in North America and South America)
Wednesday, February 24, 2021
(outside of North American and South America)
©2020 University of Waterloo
Evaluating,
Answer: (C)
Since
Note that the ones digit of a product only depends on the ones digits of the numbers being multiplied, so we could in fact multiply
Answer: (E)
The top, left and bottom unit squares each contribute 3 sides of length 1 to the perimeter.
The remaining square contributes 1 side of length 1 to the perimeter.
Therefore, the perimeter is
Alternatively, the perimeter includes 3 vertical right sides, 3 vertical left sides, 2 horizontal top sides, and 2 horizontal bottom sides, which means that the perimeter is
Answer: (A)
If
Answer: (D)
Solution 1:
10% of 500 is
100% of 500 is 500.
Thus, 110% of 500 equals
Solution 2:
110% of 500 is equal to
Answer: (E)
Since Eugene swam three times and had an average swim time of 34 minutes, he swam for
Since he swam for 30 minutes and 45 minutes on Monday and Tuesday, then on Sunday, he swam for
Answer: (C)
If
If
Therefore, if
Of the given choices, only
We can verify by direct calculation that this is the only correct answer from the given choices.
Answer: (B)
We draw an unshaded dot to represent the location of the dot when it is on the other side of the sheet of paper being shown. Therefore, the dot moves as follows:
Answer: (E)
Suppose that Janice is
Two years ago, Janice was
In 12 years, Janice will be
From the given information
Checking, if Janice is 4 years old now, then 2 years ago, she was 2 years old and in 12 years she will be 16 years old; since
Answer: (A)
Join
Since
Since
Therefore,
Since
Since
Since
Thus,
Using a similar argument, we can show that
This means that
Answer: (D)
Solution 1:
The points
This means that the distance from
Therefore, the distance from
Solution 2:
The coordinates of the 6 points are
Therefore, the distances from
Answer: (D)
Since
Since
Answer: (E)
Since
Since
Answer: (A)
Solution 1:
To find the
When the line is reflected in the
Solution 2:
When a line is reflected in the
When a line is reflected in the
Thus, when the line with equation
To find the
Answer: (D)
Amy bought and then sold
Since she bought the avocados in groups of 3, she bought
Since she paid $2 for every 3 avocados, she paid
Since she sold the avocados in groups of 5, she sold
Since she sold every 5 avocados for $4, she received
In terms of
Since we know that Amy’s profit was $100, we get
Answer: (C)
Using exponent laws,
Since
Answer: (E)
We work backwards through the given information.
At the end, there is 1 candy remaining.
Since
Thus, there were
Since
Thus, there were
Since
Thus, there were
Since
Thus, there were
Since
Thus, there were
Answer: (B)
Elina and Gustavo start by running and walking for 12 minutes.
Since there are 60 minutes in 1 hour, 12 minutes equals
When Elina runs at 12 km/h for
When Gustavo walks at 5 km/h for
At this point, Elina and Gustavo start to travel directly towards each other.
As they change direction, we use the Pythagorean Theorem to calculate their distance from each other. Using this, we obtain
Since Elina continues to travel at 12 km/h, Gustavo continues to travel at 5 km/h, and they travel directly towards each other, they close the gap at a rate of
Thus, it takes
Since there are 60 minutes in an hour, 0.153 h is equivalent to roughly 9.18 minutes.
Since Elina and Gustavo leave at 3:00 p.m. and travel for 12 minutes and then for an additional 9 minutes, they meet again at approximately 3:21 p.m.
Answer: (E)
Each of the four shaded circles has radius 1, and so has area
Next, we consider one of the three spaces. By symmetry, each of the three spaces has the same area.
Consider the leftmost of the three spaces.
Join the centres of the circles that bound this space to form a quadrilateral. Also, join the centres of these circles to the points where these circles touch the sides of the rectangle.
This quadrilateral is a square with side length 2.
The side length of this quadrilateral is 2, because each of
Next, we show that the angles of
Consider quadrilateral
Now consider quadrilateral
In a similar way, we can see that
Finally, we can consider
Similarly, the angle at
Since these angles are both
The area of the space between the four circles is thus equal to the area of the square minus the area of the four circular sectors inside the square. In fact, each of these four circular sectors is one-quarter of a circle of radius 1, since its angle at the centre of the circle is
Thus, the area of the space is equal to
This means that the total area of the shaded region equals
Of the given choices, this is closest to 15.
Answer: (D)
An integer is divisible by both 12 and 20 exactly when it is divisible by the least common multiple of 12 and 20.
The first few positive multiples of 20 are 20, 40, 60. Since 60 is divisible by 12 and neither 20 nor 40 is divisible by 12, then 60 is the least common multiple of 12 and 20.
Since
Since
This means that there are
Now, we need to remove the multiples of 60 that are also multiples of 16.
Since the least common multiple of 60 and 16 is 240, we need to remove the four-digit multiples of 240.
Since
Since
This means that there are
Finally, this means that the number of four-digit integers that are multiples of 12 and 20 but are not multiples of 16 is
Answer: (B)
We systematically work through pairs of the given integers to see which pairs add up to a third given integer. Starting with the smallest possible pairs, we have:
This means that each of the three sums in the problem must be one of the three sums above.
In the sums above, the only integer that appears three times is 27.
In the sums in the problem, the only variable that appears three times is
Therefore,
This also means that the sum
Since 12 appears in two sums and 15 does not, then
Matching the values that we know already with the equations that we have, we obtain
Answer: (C)
Suppose that the integer in the bottom left corner is
In this case, the sum of the integers in the first column is
Thus, the sum of the integers in each row, in each column, and on each diagonal also equals
Using the top row, the top right integer equals
Using the northeast diagonal, the centre integer equals
Using the second row, the middle integer in the right column equals
Using the southeast diagonal, the bottom right integer equals
Using the third row, the middle integer is
Answer: (E)
Robbie has a score of 8 and Francine has a score of 10 after two rolls each. Thus, in order for Robbie to win (that is, to have a higher total score), his third roll must be at least 3 larger than that of Francine.
If Robbie rolls 1, 2 or 3, his roll cannot be 3 larger than that of Francine.
If Robbie rolls a 4 and wins, then Francine rolls a 1.
If Robbie rolls a 5 and wins, then Francine rolls a 1 or a 2.
If Robbie rolls a 6 and wins, then Francine rolls a 1 or a 2 or a 3.
We now know the possible combinations of rolls that lead to Robbie winning, and so need to calculate the probabilities.
We recall that Robbie and Francine are rolling a special six-sided dice.
Suppose that the probability of rolling a 1 is
From the given information, the probability of rolling a 2 is
Since the combined probability of rolling a 1, 2, 3, 4, 5, or 6 equals 1, we get the equation
Thus, the probability that Robbie rolls a 4 and Francine rolls a 1 equals the product of the probabilities of each of these events, which equals
Also, the probability that Robbie rolls a 5 and Francine rolls a 1 or 2 equals
Lastly, the probability that Robbie rolls a 6 and Francine rolls a 1, a 2, or a 3 equals
Converting to the desired form, we see that
Answer: (A)
Let
We need to determine the value of
Since
This means that
Since
By the Pythagorean Theorem,
So we need to determine the values of
Join
Since
This means that
We can use the Pythagorean Theorem in
Since
Therefore,
Since
This gives the following possibilities:
1 | 1073 | 537 |
29 | 37 | 33 |
37 | 29 | 33 |
1073 | 1 | 537 |
Since
Since
This means that
Since
Answer: (C)
The distance between
Therefore, if we consider
In other words,
The slope of the line through
Therefore, this line has equation
Muliplying through by 3, we obtain
We determine the equation of the line above this line that is parallel to it and a perpendicular distance of 4 from it.
The equation of this line will be of the form
To determine the value of
To determine such a point, we draw a perpendicular of length 4 from
Since
Draw a vertical line from
Since
Since
Thus, the coordinates of
Since
In a similar way, we find that the line parallel to
This gives us the following diagram:
The points
To find the area of
The line with equation
The line with equation
The bottom triangle that is inside the square and outside
The top triangle that is inside the square and outside
Finally, this means that the area of
When we write this area in the form
Answer: (D)