Wednesday, May 11, 2016
(in North America and South America)
Thursday, May 12, 2016
(outside of North American and South America)
©2015 University of Waterloo
Evaluating,
Answer: (D)
The day on which Tanner received the most text messages will be the day with the tallest corresponding bar.
Thus, Tanner received the most text messages on Friday.
Answer: (A)
Solution 1:
A number is a multiple of 7 if it is the result of multiplying 7 by an integer.
Of the answers given, only 77 results from multiplying 7 by an integer, since
Solution 2:
A number is a multiple of 7 if the result after dividing it by 7 is an integer.
Of the answers given, only 77 results in an integer after dividing by 7, since
Answer: (C)
A positive fraction is larger than
Of the answers given,
Therefore,
Answer: (C)
Rolling the cube does not change the size of the painted triangle.
For this reason, we can eliminate answer (A).
Rolling the cube does not change the number of painted triangles.
For this reason, we can eliminate answers (D) and (E).
Rolling the cube does not change the orientation of the painted triangle with respect to the face of the cube that it is painted on.
For this reason, we can eliminate answer (C).
Of the given answers, the cube shown in (B) is the only cube which could be the same as the cube that was rolled.
Answer: (B)
The measure of the three angles in any triangle add to
Since two of the angles measure
The measure of the third angle in the triangle is
Answer: (A)
Each of the 30 pieces of fruit in the box is equally likely to be chosen. Since there are 10 oranges in the box, then the probability that the chosen fruit is an orange is
Answer: (D)
Solution 1:
Since Alex pays $2.25 to take the bus, then 20 trips on the bus would cost Alex
Since Sam pays $3.00 to take the bus, then 20 trips on the bus would cost Sam
If they each take the bus 20 times, then in total Alex would pay
Solution 2:
Since Alex pays $2.25 to take the bus, and Sam pays $3.00 to take the bus, then Alex pays
If they each take the bus 20 times, then in total Alex would pay
Answer: (C)
Solution 1:
Travelling at a constant speed of 85 km/h, the entire 510 km trip would take Carrie
Since Carrie is halfway through the 510 km trip, then the remainder of the trip will take her half of the total trip time or
Solution 2:
Carrie is halfway through a 510 km trip, and so she has half of the distance or
Since Carrie travels at a constant speed of 85 km/h, then it will take her
Answer: (E)
Since
The distance between
Since
That is,
Answer: (A)
In the diagram, there are 4 rows of octagons and each row contains 5 octagons.
Therefore, the total number of octagons in the diagram is
In the diagram, there are 3 rows of squares and each row contains 4 squares.
Therefore, the total number of squares in the diagram is
The ratio of the number of octagons to the number of squares is
Answer: (E)
The sum of the units column is
Since
Then
The sum of the tens column becomes
Since
Then
We may verify that the sum of the hundreds column is
The value of
Answer: (B)
Since a cube is a rectangular prism, its volume is equal to the area of its base,
A cube has edges of equal length and so
Thus, the volume of a cube is the product of three equal numbers.
The volume of the larger cube is 64 cm
The smaller cube has edges that are half the length of the edges of the larger cube, or 2 cm.
The volume of the smaller cube is
Answer: (C)
Ahmed could choose from the following pairs of snacks: apple and orange, apple and banana, apple and granola bar, orange and banana, orange and granola bar, or banana and granola bar.
Therefore, there are 6 different pairs of snacks that Ahmed may choose.
Answer: (D)
Sophia did push-ups for 7 days (an odd number of days), and on each day she did an equal number of push-ups more than the day before (5 more).
Therefore, the number of push-ups that Sophia did on the middle day (day 4) is equal to the average number of push-ups that she completed each day.
Sophia did 175 push-ups in total over the 7 days, and thus on average she did
Therefore, on day 4 Sophia did 25 push-ups, and so on day 5 she did
(Note: We can check that
Answer: (E)
Since
Since
(Can you explain why each of the other answers is not equal to
Answer: (B)
Each of the following four diagrams shows the image of triangle
Thus, each of the triangles labelled
The triangle labelled
Answer: (C)
The mean (average) of the set of six numbers is 10, and so the sum of the set of six numbers is
If the number 25 is removed from the set, the sum of the set of the remaining five numbers is
The mean (average) of the remaining set of five numbers is
Answer: (B)
The shaded and unshaded sections of the ribbon have equal length.
Since there are 5 such sections, then each shaded and unshaded section has length equal to
All measurements which follow are made beginning from the left end of the ribbon.
Point
Point
All points are equally spaced, and so points
Since
Thus, if Suzy makes a vertical cut at point
We note that no point is located more than 2 sections from the right end of the ribbon.
That is, no point is located more than
Answer: (C)
We begin by naming the boxes as shown.
Of the five answers given, the integer which cannot appear in box
Since boxes
Since boxes
Next, we consider the possibilities if 20 is to appear in box
If 3 appears in box
However, there are no two integers from 1 to 9 whose product is 60 and so there are no possible integers which could be placed in boxes
If any integer greater than or equal to 4 appears in box
However, the maximum value that can appear in box
Therefore, there are no possible integers from 1 to 9 which can be placed in boxes
The diagrams below demonstrate how each of the other four answers can appear in box
Answer: (D)
Line segment
This column contains 9 points other than
Line segment
This row contains 9 points other than
Each of these 9 points is different from the 9 points in the column containing
Thus, there are
Since there are a total of 99 points to choose
Answer: (A)
First we choose to label one of the vertices 2, and then label the vertex that is farthest away from this vertex
(Can you explain why the vertex labelled
Each of the other six vertices of the cube lie on one of the three faces on which the vertex labelled 2 lies.
We note that the vertex labelled
From the six given lists, we consider those lists in which the number 2 appears.
These are:
Thus, the vertices labelled
The only vertex label not included in this list is 6.
Thus, the vertex labelled 6 is the only vertex which does not lie on a face on which the vertex labelled 2 lies.
Therefore, the correct labelling for vertex
One possible labelling of the cube is shown.
Answer: (D)
Solution 1:
Let the letter
On her first draw, Angie may draw
Case 1: Angie draws
If Angie draws
On her second draw, Angie may draw
If she draws
Since there are no red marbles remaining, it is not possible for the final marble to be red in this case.
If on her second draw Angie instead draws
When these are both drawn on her third draw, the
Again in this case it is not possible for the final marble to be red.
Thus, if Angie draws
Case 2: Angie draws
If Angie draws
On her second draw, Angie may draw
If she draws
When these are both drawn on her third draw, the
In this case it is not possible for the final marble to be red.
Thus, if Angie draws
Therefore, under the given conditions of drawing and discarding marbles, the probability that Angie’s last remaining marble is red is zero.
Solution 2:
Let the letter
If the final remaining marble is
That is, the last two marbles must be
If the last two marbles are
Thus it is not possible for the final marble to be
So the final remaining marble is
If the final two marbles are
However, if the final three marbles are
That is, it is not possible for the final two marbles in the jar to be
The only possibility that the final remaining marble is
Therefore, under the given conditions of drawing and discarding marbles, the probability that Angie’s last remaining marble is red is zero.
Answer: (E)
We begin by showing that each of
Now,
Why is this? Suppose that
Since
Since the numbers in the list are equally spaced, then
(For example, the average of the 5 integers
But the sum of the integers equals the average of the integers (
Now
Therefore,
Thus,
Further,
Why is this? Suppose that
Since
Since the numbers in the list are equally spaced, then the average of the numbers in the list is the average of
(For example, the average of the 6 integers
But the sum of the integers equals the average of the integers (
Now
Therefore,
Thus,
Therefore,
A similar argument shows that every power of 2 cannot be written as the sum of any number of consecutive positive integers.
Returning to the original question, exactly one of the five numbers in the original list cannot be written in the desired way, and so the answer is (B).
Answer: (B)
Consider the diagonal lines that begin on the left edge of the triangle and move downward to the right.
The first number in the
For example, the first number in the
The second number in the
The third number in the
Following this pattern, the
The table below demonstrates this for
Horizontal Row Number | ||
---|---|---|
1 | 3 | 3 |
2 | ||
3 | ||
4 | ||
5 | ||
⋮ | ⋮ | ⋮ |
The number 2016 lies in some diagonal line(s).
To determine which diagonal lines 2016 lies in, we express 2016 as a product
Further, if
We want the horizontal row in which 2016 first appears, and so we must find positive integers
In the table below, we summarize the factor pairs
Factor Pair |
Horizontal Row Number |
---|---|
2016 | |
1009 | |
674 | |
507 | |
341 | |
294 | |
259 | |
232 | |
179 | |
157 | |
141 | |
129 | |
116 | |
107 | |
99 | |
94 | |
91 | |
89 |
(Note: By recognizing that when
We have included all possible pairs
We see that 2016 will appear in 18 different locations in the triangle.
However, the first appearance of 2016 occurs in the horizontal row numbered 89.
Answer: (E)
Evaluating,
Answer: (A)
Solution 1:
The fraction
Solution 2:
Since
Answer: (B)
Reading from the graph, we summarize the number of hours that Stan worked each day in the table below.
Day | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|
Number of Hours | 2 | 0 | 3 | 1 | 2 |
Therefore, Stan worked a total of
Answer: (C)
Written numerically, three tenths plus four thousandths is
Answer: (C)
Folds occur along the five edges between adjoining faces in the figure shown.
Consider the face numbered 3 as being the bottom face of the completed cube.
First, fold upward along the four edges of the face numbered 3 (the edges between 3 and 5, 3 and 4, 3 and 6, and 3 and 2).
After folding upward, the faces numbered
The final fold occurs along the edge between the faces numbered 1 and 2.
The face numbered 1 becomes the top face of the cube after this fold.
Since the bottom face is opposite the top face, then the face numbered 3 is opposite the face numbered 1.
Answer: (B)
Side
Side
Therefore, the coordinates of
Answer: (D)
A rectangle with a width of 2 cm and a length of 18 cm has area
The area of a square with side length
The area of the square is also 36 cm
Answer: (A)
From the list
The numbers
The ratio of the number of prime numbers to the number of composite numbers is
Answer: (A)
Since 10% of 200 is
Answer: (C)
The circumference of a circle with radius
When
Answer: (C)
In equilateral triangle
Since
In isosceles triangle
Therefore,
Answer: (E)
We try each of the five options:
(A):
(B):
(C):
(D):
(E):
Therefore, the correct operations are, in order,
Answer: (E)
Ahmed could choose from the following pairs of snacks: apple and orange, apple and banana, apple and granola bar, orange and banana, orange and granola bar, or banana and granola bar.
Therefore, there are 6 different pairs of snacks that Ahmed may choose.
Answer: (D)
One soccer ball and one soccer shirt together cost $100.
So then two soccer balls and two soccer shirts together cost
Since we are given that two soccer balls and three soccer shirts together cost $262, then $200 added to the cost of one soccer shirt is $262.
Thus, the cost of one soccer shirt is
Answer: (A)
The map’s scale of
So then 2 cm measured on the map represents an actual distance of
The actual distance between Gausstown and Piville is 12 km.
Answer: (A)
The mean (average) of the set of six numbers is 10, and so the sum of the set of six numbers is
If the number 25 is removed from the set, the sum of the set of the remaining five numbers is
The mean (average) of the remaining set of five numbers is
Answer: (B)
The positive integers between 10 and 2016 which have all of their digits the same are:
To be divisible by 3, the sum of the digits of the positive integer must equal a multiple of 3.
From the list above, the only 2-digit numbers whose digit sum is a multiple of 3 are
(We may verify that each of the other digit sums,
A 3-digit positive integer with all digits equal to
Thus, all 3-digit positive integers with equal digits are divisible by 3.
That is, all 9 of the 3-digit integers listed above are divisible by 3.
Finally, the number 1111 has digit sum 4 and thus is not divisible by 3.
There are
Answer: (B)
Joe used
Since Joe used
If Joe has already travelled 165 km, then he can travel another
Answer: (E)
The first scale shows that 2
Thus, we may eliminate answer (C).
The second scale shows that 2
Thus, we may eliminate answer (B).
Since 1
Thus, we may eliminate answer (E).
Since 1
Thus, we may eliminate answer (A).
Finally, we are left with answer (D).
Since 1
Thus, answer (D) is the only answer which is not true.
Answer: (D)
Points
Points
That is, in
Using the Pythagorean Theorem,
Answer: (A)
If the ten thousands digits of the two numbers differ by more than 1, then the two numbers will differ by more than 10 000. (For example, a number of the form
Since all of the given answers are less than 1000 and since the two ten thousands digits cannot be equal, then the ten thousands digits must differ by 1. We will determine the exact ten thousands digits later, so we let the smaller of the two ten thousands digits be
To make the difference between
In other words, we try to make
To make
Since all of the digits must be different, then the minimum possible value of
To make
Since all of the digits must be different, then the maximum possible value of
Since we have made
The digits that have not been used are
This gives numbers
Their difference is
Answer: (C)
We find the area of the shaded region by determining the area of the unshaded region and subtracting this from the total area of the rectangle.
We begin by extending
Since
Since
Since
Also,
Since
Since
Since
Therefore,
Since
In quadrilateral
That is, quadrilateral
The area of trapezoid
The total area of the shaded region is found by subtracting the area of
The area of rectangle
Answer: (D)
For Zeus to arrive at the point
More specifically, Zeus will need to make at least 1056 moves to the right (
Since Zeus cannot move in the same direction twice in a row, then no two moves
Since there are 1056 moves
Therefore, at least
We will show that there is actually a sequence of 2111 moves that obey the given rules and that take Zeus to
For Zeus to end at a point with
So we put one
This leaves us with
We want to fill these spaces with moves that do not affect Zeus’ position (since we already have him moving 1056 moves
We can do this by making the first
We have shown that Zeus needs at least 2111 moves to get to
Answer: (D)
When two integers are multiplied together, the final two digits (the tens digit and the units digits) of the product are determined by the final two digits of each of the two numbers that are multiplied.
This is true since the place value of any digit contributes to its equal place value (and possibly also to a greater place value) in the product.
That is, the hundreds digit of each number being multiplied contributes to the hundreds digit (and possibly to digits of higher place value) in the product.
Thus, to determine the tens digit of any product, we need only consider the tens digits and the units digits of each of the two numbers that are being multiplied.
For example, to determine the final two digits of the product
Since
Since the final two digits of
Then
Further,
Since
This tells us that
Finally,
Answer: (B)
We begin by adding variables to some of the blanks in the grid to make it easier to refer to specific entries:
Since the numbers in each row form an arithmetic sequence and the numbers in each column form an arithmetic sequence, we will refer in several sequences to the common difference.
Let the common difference between adjacent numbers as we move down column 3 be
Therefore,
Also,
Moving from left to right along row 2, the common difference between adjacent numbers must be
Therefore,
Moving from the top to the bottom of column 5, the common difference between adjacent numbers can be found by subtracting
That is, the common difference between adjacent numbers in column 5 is
Moving down column 5, we can see that
This gives the following updated grid:
In other words,
We note that
Therefore,
Similarly,
Therefore, the sum of the digits of the value of
Answer: (B)