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: (D)
Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square.
Thus, the side length of a square with perimeter 28 is
Answer: (E)
In the diagram, there are 9 hexagons of which 5 are shaded.
Therefore, the fraction of all of the hexagons that are shaded is
Answer: (B)
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)
We know that
Answer: (D)
Since
Dividing both sides of this equation by 2, we obtain
Answer: (B)
The segment of the number line between 3 and 33 has length
Since this segment is divided into six equal parts, then each part has length
The segment
The segment
Thus, the sum of the lengths of
Answer: (A)
Since
We note that
This means that both
In other words, of the five numbers
Since the list contains 5 numbers, then its median is the third largest number, which is 2019.
(Note that it does not matter whether
Answer: (D)
Since the complete angle at the centre of each circle is
In other words, each of the circles is
There are 12 circles in the diagram.
Since the radius of each circle is 1, then the area of each circle is
Therefore, the total shaded area is
Answer: (D)
Suppose that sixty
Each of the 60 cubes has its front, top and back faces exposed.
The leftmost and rightmost cubes also have their left and right faces, respectively, exposed.
No other faces are exposed.
Therefore, the number of
Answer: (C)
Using the second row, we see that the sum of the numbers in each row, column and diagonal must be
Since the sum of the numbers in the first column must be 9, then the bottom left number must be
Since the sum of the numbers in the top left to bottom right diagonal must be 9, then the bottom right number must be
Since the sum of the numbers in the bottom row must be 9, then
We can complete the magic square as shown:
Answer: (E)
Since
Since
Since the angles in any triangle have a sum of
Answer: (C)
Solution 1
Since
This means that
Solution 2
Since
Since
This means that
Answer: (B)
Since the ratio of the number of skateboards to the number of bicycles was
Since the difference between the numbers of skateboards and bicycles is 12, then
Therefore, the total number of skateboards and bicycles is
Answer: (A)
For Sophie’s average over 5 tests to be 80%, the sum of her marks on the 5 tests must be
After the first 3 tests, the sum of her marks is
Therefore, she will reach her goal as long as the sum of her marks on the two remaining tests is at least
The sums of the pairs of marks given are (A) 161%, (B) 161%, (C) 162%, (D) 156%, (E) 160%.
Thus, the pair with which Sophie would not meet her goal is (D).
Answer: (D)
Solution 1
Since the result must be the same for any real number
(A)
(B) (C) (D) (E)
Therefore,
Solution 2
For any real number
Therefore, neither
For any negative real number
Therefore,
Thus, the least of the five values is either
When
Since the difference between
Answer: (E)
Each of the animals is either striped or spotted, but not both.
Since there are 100 animals and 62 are spotted, then there are
Each striped animal must have wings or a horn, but not both.
Since there are 28 striped animals with wings, then there are
Each animal with a horn must be either striped or spotted.
Since there are 36 animals with horns, then there are
Answer: (E)
By the Pythagorean Theorem,
Since
By the Pythagorean Theorem,
Since
Since
Since
Since
Since the sum of the three areas is
Since
Answer: (C)
Since 4 balls are chosen from 6 red balls and 3 green balls, then the 4 balls could include:
4 red balls, or
3 red balls and 1 green ball, or
2 red balls and 2 green balls, or
1 red ball and 3 green balls.
There is only 1 different-looking way to arrange 4 red balls.
There are 4 different-looking ways to arrange 3 red balls and 1 green ball: the green ball can be in the 1st, 2nd, 3rd, or 4th position.
There are 6 different-looking ways to arrange 2 red balls and 2 green balls: the red balls can be in the 1st/2nd, 1st/3rd, 1st/4th, 2nd/3rd, 2nd/4th, or 3rd/4th positions.
There are 4 different-looking ways to arrange 1 red ball and 3 green balls: the red ball can be in the 1st, 2nd, 3rd, or 4th position.
In total, there are
Answer: (A)
Since the sides of quadrilateral
This means that quadrilateral
Since the diagram does not change when rotated by
We calculate the area of
Extend
Since
Since
Since
Since
Since the sides of
Since
Thus,
Answer: (B)
The units digit of
This is because the units digit of any power of 5 is 5.
To see this, we note that the first few powers of 5 are
The units digit of
This is because the units digits of powers of 3 cycle
To see this, we note that the first few powers of 3 are
Since the units digits of powers of 3 cycle in groups of 4 and
Moving three additional positions along the sequence, the units digit of
Since the units digit of
Answer: (E)
Solution 1
The smallest integer greater than 2019 that can be formed in this way is formed using the next two largest consecutive integers 20 and 21, giving the four-digit integer 2120.
The largest such integer is 9998.
The list of such integers is
Since the numbers in the list are equally spaced, then their sum will equal the number of numbers in the list times the average number in the list.
The average number in the list is
Since each number in the list is 101 greater than the previous number, then the number of increments of 101 from the first number to the last is
Since the number of increments is
This means that the sum of the numbers in the list is
Solution 2
As in Solution 1, the list of such integers is
If the sum of these integers is
Therefore,
Answer: (C)
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 label 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
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.
Therefore, the first shaded section touches shaded stripes between 1 m and 2 m, and between 3 m and
The second shaded section touches shaded stripes between 7 m and 8 m, and between 9 m and
Therefore, the total length of “shaded on shaded” 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 20%, or choice (A).
Answer: (A)
First, we note that
Therefore,
Since
The smallest integer of the form
Since
Suppose that
Comparing units digits, we see that the units digit of
This means that
In the first case,
This integer has a units digit of 8.
For this integer to have a tens digit of 4, we need
This means that
This means that
Since
In the second case,
This integer has a units digit of 8.
For this integer to have a tens digit of 4, we need
This means that
This means that
Since
In total, there are
Answer: (E)
In
This is because
Let the area of
We know that
Since
Finally, looking at points
Therefore,
Of the given choices, the area of
Answer: (E)