Tuesday, February 28, 2017
(in North America and South America)
Wednesday, March 1, 2017
(outside of North American and South America)
©2016 University of Waterloo
Evaluating,
Answer: (A)
In each of the 6 rows, there is 1 unshaded square, and so there are
Since there are 6 rows, then there are
Alternatively, we note that there are
Answer: (B)
In the diagram, there are 5 shaded and 3 unshaded triangles, and so the ratio of the number of shaded triangles to the number of unshaded triangles is
Answer: (B)
We note that
This means that
Since
Answer: (C)
We need to determine the time that is 30 hours after 2 p.m. on Friday.
The time that is 24 hours after 2 p.m. on Friday is 2 p.m. on Saturday.
The time that is 30 hours after 2 p.m. on Friday is an additional 6 hours later.
This time is 8 p.m. on Saturday.
Answer: (E)
The time period in which the number of people at the zoo had the largest increase is the time period over which the height of the bars in the graph increases by the largest amount.
Looking at the graph, this is between 11:00 a.m. and 12:00 p.m.
(We note that the first three bars represent numbers between 200 and 400, and the last three bars represent numbers between 600 and 800, so the time period between 11:00 a.m. and 12:00 p.m. is the only one in which the increase was larger than 200.)
Answer: (C)
Since
(We did not have to determine the value of
Answer: (D)
The three integers from the list whose product is 80 are 1, 4 and 20, since
The sum of these integers is
(Since 80 is a multiple of 5 and 20 is the only integer in the list that is a multiple of 5, then 20 must be included in the product. This leaves two integers to choose, and their product must be
Answer: (C)
Since Jovin, Anna and Olivia take
Answer: (B)
When
When
Only the fifth expression (
We note further that since 2017 is an odd integer and
Answer: (E)
When Ursula runs 30 km at 10 km/h, it takes her
This means that Jeff completes the same distance in
Therefore, Jeff’s constant speed is
Answer: (D)
Since the area of the larger square equals the sum of the areas of the shaded and unshaded regions inside, then the area of the larger square equals
Since the larger square has an area of
Answer: (C)
Solution 1
We undo Janet’s steps to find the initial number.
To do this, we start with 28, add 4 (to get 32), then divide the sum by 2 (to get 16), then subtract 7 (to get 9).
Thus, Janet’s initial number was 9.
Solution 2
Let Janet’s initial number be
When she added 7 to her initial number, she obtained
When she multiplied this sum by 2, she obtained
When she subtracted 4 from this result, she obtained
Since her final result was 28, then
Answer: (A)
Since the tax rate is 10%, then the tax on each $2.00 app is
Therefore, including tax, each app costs
Since Tobias spends $52.80 on apps, he downloads
Therefore,
Answer: (D)
Let
The sum of the heights of the squares on the right side is
The sum of the heights of the squares on the left side is
Since the two sums are equal, then
Therefore, the square with area
In other words,
Answer: (C)
The six angles around the centre of the spinner add to
Thus,
Therefore, the sum of the central angles of the shaded regions is
The probability that the spinner lands on a shaded region is the fraction of the entire central angle that is shaded, which equals the sum of the central angles of the shaded regions divided by the total central angle (
Answer: (A)
Since Igor is shorter than Jie, then Igor cannot be the tallest.
Since Faye is taller than Goa, then Goa cannot be the tallest.
Since Jie is taller than Faye, then Faye cannot be the tallest.
Since Han is shorter than Goa, then Han cannot be the tallest.
The only person of the five who has not been eliminated is Jie, who must thus be the tallest.
Answer: (E)
From the number line shown, we see that
If
Since this is not the case, then it is not true that
If
If
Therefore, it must be the case that
If
Therefore, it must be the case that
From the given possibilities, this means that
We can check that if
Answer: (C)
Since
Since the angles in
Since
But
Since
This means that
Therefore,
Answer: (A)
We call the
In other words, these are the unit cubes none of whose faces form a part of any of the faces of the large cube.
These unit cubes form a cube that is
To see why this is true, imagine placing the original painted large cube on a table.
Each unit cube with at least one face that forms part of one of the outer faces (or outer layers) has paint on at least one face.
First, we remove the top and bottom layers of unit cubes. This creates a rectangular prism that is
Next, we can remove the left, right, front, and back faces.
This leaves a cube that is
Therefore,
The unit cubes that have exactly 1 gold face are those unit cubes that are on the outer faces of the large cube but do not touch the edges of the large cube.
Consider each of the six
The unit cubes that have 1 gold face are those with at least one face that forms part of a face of the large cube, but do not share any edges with the edges of the large cube. Using a similar argument to above, we can see that these unit cubes form a
There are thus
We calculate the values of
Choice | |||
---|---|---|---|
(A) | |||
(B) | |||
(C) | |||
(D) | |||
(E) |
Answer: (C)
The averages of groups of three numbers are equal if the sums of the numbers in each group are equal, because in each case the average equals the sum of the three numbers divided by 3.
Therefore, the averages of three groups of three numbers are equal if the sum of each of the three groups are equal.
The original nine numbers have a sum of
Consider the middle three numbers. Since two of the numbers are 13 and 17, then the third number must be
Therefore, the number that is placed in the shaded circle is
Answer: (D)
The perimeter of
We know that
Let
Since
We join
Since
Since
This means that the radius of the semi-circle is 10, and so
Now
By the Pythagorean Theorem,
Each of
Therefore, we can apply the Pythagorean Theorem again to obtain
Since
Therefore, the perimeter of
Of the given choices, this is closest to 86.
Answer: (B)
The squares of the one-digit positive integers
Of these, the squares
In other words,
Thus,
To find all even two-digit Anderson numbers, we note that any two-digit even Anderson number
So we need to look for two-digit Anderson numbers
Another way of writing the number
In this case,
Note that
For
Thus, the tens digit of
This means that
When we check the nine possible values for
This means that
Note that
Next, we look for three-digit even Anderson numbers
Using a similar argument to above, we see that
In other words,
In this case,
We note that the tens and units digits of
Now
Thus,
Again, checking the nine possible values for
This means that
Note that
Since Anderson numbers are less than 10 000, then we still need to look for four-digit even Anderson numbers.
Again, using a similar argument, we see that
In other words,
In this case,
We note that the hundreds, tens and units digits of
Now
Thus,
Again, checking the nine possible values for
This means that
Note that
Thus,
The sum of the digits of
Answer: (E)
Since there are 1182 houses that have a turtle, then there cannot be more than 1182 houses that have a dog, a cat, and a turtle.
Since there are more houses with dogs and more houses with cats than there are with turtles, it is possible that all 1182 houses that have a turtle also have a dog and a cat.
Therefore, the maximum possible number of houses that have all three animals is 1182, and so
Since there are 1182 houses that have a turtle and there are 2017 houses in total, then there are
Now, there are 1651 houses that have a cat.
Since there are 835 houses that do not have a turtle, then there are at most 835 houses that have a cat and do not have a turtle. In other words, not all of the houses that do not have a turtle necessarily have a cat.
This means that there are at least
Lastly, there are 1820 houses that have a dog.
Since there are at least 816 houses that have both a cat and a turtle, then there are at most
Since there are 1820 houses that do have a dog, then there are at least
In other words, the minimum possible number of houses that have all three animals is 619, and so
The two Venn diagrams below show that each of these situations is actually possible:
Since
Answer: (C)
We label the digits of the unknown number as
Since
Since
Since
Case 1:
Since
If
Thus,
To this point, this form is consistent with the 1st, 2nd, 3rd and 7th rows of the table.
Since
Since
Since
But we already know that
Therefore,
We note that the integers
Case 2:
Since
Since
Since
This means that we must have
Thus,
To this point, this form is consistent with the 1st, 2nd, 3rd, 4th, and 7th rows of the table.
Since
Since
In this case, the integer 39542 is the only possibility, and it satisfies all of the requirements.
Case 3:
Since
But if
Thus,
To this point, this form is consistent with the 1st, 2nd, 3rd and 7th rows of the table.
Since
Since
Since
If
Thus, it must be the case that
Therefore,
We note that the integers
Thus, there are 13 possibilities for Sam’s numbers and the sum of these is
Answer: (E)