Tuesday, February 26, 2019
(in North America and South America)
Wednesday, February 27, 2019
(outside of North American and South America)
©2018 University of Waterloo
Evaluating,
Answer: (A)
Kai was born 25 years before 2020 and so was born in the year
Answer: (C)
Since 38% of students received a muffin, then
Alternatively, using the percentages of students who received yogurt, fruit or a granola bar, we see that
Answer: (D)
Re-arranging the order of the numbers being multiplied,
Answer: (C)
Since
Answer: (A)
The line with equation
When the line is translated 6 units downwards, all points on the line are translated 6 units down.
This moves the
Answer: (E)
Since the average of
Multiplying by 3, we obtain
Re-arranging, we obtain
Answer: (E)
To get from
To get from
To get from
To get from
To get from
Therefore, the shortest path is from
Answer: (E)
Since
Thus,
Using the given answers,
Answer: (C)
Matilda and Ellie each take
Matilda paints
Ellie paints
Therefore,
Answer: (A)
We let the values in the two unlabelled circles be
From the given rules,
Also,
Finally,
Answer: (B)
Join
Since
Since
Since
Answer: (A)
From the ones column, we see that
Since
Since its ones digit is 2, then
This also means that there is a carry of 1 into the tens column.
From the tens column, we see that
Since
Since its ones digit is 4, then
This also means that there is a carry of 2 into the hundreds column.
From the hundreds column, we see that
Since
Since its ones digit is 0, then
This also means that there is a carry of 2 into the thousands column.
This means that
This gives
Answer: (B)
Each letter A, B, C, D, E appears exactly once in each column and each row.
The entry in the first column, second row cannot be A or E or B (the entries already present in that column) and cannot be C or A (the entries already present in that row).
Therefore, the entry in the first column, second row must be D.
This means that the entry in the first column, fourth row must be C.
The entry in the fifth column, second row cannot be D or C or A or E and so must be B.
This means that the entry in the second column, second row must be E.
Using similar arguments, the entries in the first row, third and fourth columns must be D and B, respectively.
This means that the entry in the second column, first row must be C.
Using similar arguments, the entries in the fifth row, second column must be A.
Also, the entry in the third row, second column must be D.
This means that the letter that goes in the square marked with
We can complete the grid as follows:
Answer: (B)
The slope of line segment
Since
This means that the slopes of
Since the slope of
Since the “run” of
Thus,
Answer: (C)
Suppose that there are
This means that there are
Including Kaukab, the total number of people in line is
Of the given choices (23, 20, 24, 21, 25), the only one that is one more than a multiple of 3 is 25, which equals
Therefore, a possible value for
Answer: (E)
Consider the triangular-based prism on the front of the rectangular prism.
This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back.
The rectangle on the front measures
The rectangle on the left measures
The triangles on the top and bottom each are right-angled and have legs of length 5 and 12. This means that each has area
The rectangle on the back has height 3. The length of this rectangle is the length of the diagonal of the bottom face of the rectangular prism. By the Pythagorean Theorem, this length is
In total, the surface area of the triangular prism is thus
Answer: (D)
André runs for 10 seconds at a speed of
Therefore, André runs
Carl runs for 20 seconds before André starts to run and then 10 seconds while André is running. Thus, Carl runs for 30 seconds.
Since Carl runs at a speed of
Since André and Carl run the same distance, then
Thus,
Answer: (D)
Using exponent laws, the expression
Since
The corresponding values of
Therefore, the sum of the possible values of
Answer: (A)
Let the radii of the circles with centres
The distance between the centres of two touching circles equals the sum of the radii of these circles.
Therefore,
Also,
Adding these three equations, we obtain
Since
Since
Since
Knowing the radii of the circles will allow us to calculate the dimensions of the rectangle.
The height of rectangle
Thus, the height of rectangle
To calculate the width of rectangle
Since radii are perpendicular to tangents at points of tangency, then
Each of
Thus,
By a similar argument,
Thus,
Since
By the Pythagorean Theorem,
This means that
Therefore, the area of rectangle
Of the given choices, this answer is closest to (E) 3950.
Answer: (E)
Solution 1
We start with the ones digits.
Since
Looking at the tens column, since
Looking at the hundreds column, since
Looking at the thousands column, since
Looking at the ten thousands column, since
Looking at the hundred thousands column,
This gives the following completed multiplication:
Solution 2
Let
This means that
Also,
From the given multiplication,
Thus,
Since
Answer: (A)
Here is one way in which the seven friends can ride on four buses so that the seven restrictions are satisfied:
Bus 1 | Bus 2 | Bus 3 | Bus 4 |
---|---|---|---|
Abu | Bai | Don | Gia |
Cha | Fan | Eva |
At least 3 buses are needed because of the groups of 3 friends who must all be on different buses.
We will now show that it is impossible for the 7 friends to travel on only 3 buses.
Suppose that the seven friends could be put on 3 buses.
Since Abu, Bai and Don are on 3 different buses, then we assign them to three buses that we can call Bus 1, Bus 2 and Bus 3, respectively. (See the table below.)
Since Abu, Bai and Eva are on 3 different buses, then Eva must be on Bus 3.
Since Cha and Bai are on 2 different buses and Cha and Eva are on 2 different buses, then Cha cannot be on Bus 2 or Bus 3, so Cha is on Bus 1.
So far, this gives
Bus 1 | Bus 2 | Bus 3 |
---|---|---|
Abu | Bai | Don |
Cha | Eva |
The remaining two friends are Fan and Gia.
Since Fan, Cha and Gia are on 3 different buses, then neither Fan nor Gia is on Bus 1.
Since Don, Gia and Fan are on 3 different buses, then neither Fan nor Gia is on Bus 3.
Since Gia and Fan are on separate buses, they cannot both be on Bus 2, which means that the seven friends cannot be on 3 buses only.
Therefore, the minimum number of buses needed is 4.
Answer: (B)
Since the wheel turns at a constant speed, then the percentage of time when a shaded part of the wheel touches a shaded part of the path will equal the percentange of the total length of the path where there is “shaded on shaded” contact.
Since the wheel has radius 2 m, then its circumference is
Since the wheel is divided into four quarters, then the portion of the circumference taken by each quarter is
We call the left-hand end of the path 0 m.
As the wheel rotates once, the first shaded section of the wheel touches the path between 0 m and
As the wheel continues to rotate, the second shaded section of the wheel touches the path between
While the wheel makes 3 complete rotations, a shaded quarter will be in contact with the path over 6 intervals (2 intervals per rotation).
The path is shaded for 1 m starting at each odd multiple of 1 m, and unshaded for 1 m starting at each even multiple of 1 m.
We make a chart of the sections where shaded quarters touch the path and the parts of these intervals that are shaded:
Beginning of quarter (m) | End of quarter (m) | Shaded parts of path (m) |
---|---|---|
1 to 2; 3 to |
||
7 to 8; 9 to |
||
13 to 14; 15 to |
||
19 to 20; 21 to |
||
Therefore, the total length of “shaded on shaded”, in metres, is
The total length of the path along which the wheel rolls is
This means that the required percentage of time equals
Of the given choices, this is closest to 26%, or choice (E).
Answer: (E)
We let
First, we note that the integer
This means that the product
This means that the number of possible values for
We note that
We count the number of possible values for
Let
Case 1:
If
This means that there is only one possible value for
Case 2:
Here, it must be the case that
Case 3:
Here, it must be the case that two of
In other words,
This means that there is only one possible value for
Case 4:
Here,
This means that there is only one possible value for
Case 5:
Here, two of
This means that the possible values for the third of these are
This means that there are 6 possible values for
Case 6:
As in Case 5, there are 6 possible values for
Case 7:
Here, one of
This means that there are 6 possible values for
Case 8:
Here, one of
Each of
We make a multiplication table to determine the possible values of
Therefore, there are 16 possible values of
Case 9:
As in Case 8, there are 16 possible values of
Case 10:
Here, none of
This means that each of
This means that the only possible prime factors of
Each of
This means that
Let
If
If
If
Thus,
There is duplication in this list and so the values of
If
In the first situation,
These can be written as
In the second situation,
These equal
Combining lists, we get
Therefore, in this case there are 5 possible values for
If
In the first situation, the other two of
Note that
Thus
If two of
Thus,
Combining these possibilities,
If
Thus,
Each of the number of factors of 2 between 2 and 7, inclusive, is possible, so there are 6 possible values of
If
Thus, each of
Thus,
In total, the number of possible values of
Answer: (A)
Suppose that
Since
By the Pythagorean Theorem,
By the Pythagorean Theorem,
By the Pythagorean Theorem,
Adding these three equations, we obtain
Since
Since
Since
Since the right side of this last inequality is negative and the left side is non-negative, then this inequality is always true.
Therefore, it must be true that
Since all three parts of this inequality are positive, then
Since
The number of integers
Every such value of
Therefore, there are 1981 such integers
Answer: (E)