Wednesday, February 23, 2022
(in North America and South America)
Thursday, February 24, 2022
(outside of North American and South America)
©2021 University of Waterloo
Evaluating,
Answer: (D)
From the graph, we see that Haofei donated $2, Mike donated $6,
Pierre donated $2, and Ritika donated $8.
In total, the four students donated
Answer: (B)
In the given sum, each of the four fractions is equivalent to
Therefore, the given sum is equal to
Answer: (E)
On a number line,
This means that
If we start at
It then takes an additional 0.6 units to move in the negative direction
from
Therefore,
Alternatively, when comparing
Answer: (B)
From the diagram,
Answer: (A)
Between them, Robyn and Sasha have
If each does the same number of tasks, each must do
This means that Robyn must do
Answer: (C)
Because all of the angles in the figure are right angles, each
line segment is either horizontal or vertical.
The height of the figure is
This means that the length of the unmarked vertical segment must equal
Also, the length of the unmarked horizontal segment must equal
Starting in the top left corner and adding lengths in a clockwise
direction, the perimeter is
Alternatively, we can "complete the rectangle" by sliding the shortest
horizontal side and the shortest vertical side as shown to form a
rectangle with height
The perimeter of this rectangle is
Answer: (E)
The total central angle in a circle is
Since the Green section has an angle at the centre of the circle of
This means that when the spinner is spun once, the probability that it
lands on the Green section is
Similarly, the probability that the spinner lands on Blue is also
Since the spinner lands on one of the four colours, the probability that
the spinner lands on either Red or Yellow is
Answer: (D)
Since the line with equation
Substituting
Answer: (E)
We label Mathville as
There is 1 route from
There are 3 routes to
This means that there are 4 routes to
Answer: (C)
Since the given grid is
This means that
Join
Since
We note further that
Since
Similarly,
Thus, the perimeter of
Answer: (C)
The integers between 1 and 100 that have a ones digit equal to 6
are
The additional integers between 1 and 100 that have a tens digits equal
to 6 are
Since the digit 6 must occur as either the ones digit or the tens digit,
there are
Answer: (C)
Suppose that Rosie runs
Since Mayar runs twice as fast as Rosie, then Mayar runs
When Mayar and Rosie meet, they will have run a total of 90 m, since
between the two of them, they have covered the full 90 m.
Therefore,
Since
Answer: (D)
We use
We use the notation
The five sentences give
This means that two people are older than Bev and two people are younger
than Bev, which means that Bev must be the third oldest.
Answer: (B)
We note that all of the given possible sums are odd, and also
that every prime number is odd with the exception of 2 (which is
even).
When two odd integers are added, their sum is even.
When two even integers are added, their sum is even.
When one even integer and one odd integer are added, their sum is
odd.
Therefore, if the sum of two integers is odd, it must be the sum of an
even integer and an odd integer.
Since the only even prime number is 2, then for an odd integer to be the
sum of two prime numbers, it must be the sum of 2 and another prime
number.
Note that
Answer: (A)
Since 60 games are played and each of the 3 pairs plays the same
number of games, each pair plays
Alvin wins 20% of the 20 games that Alvin and Bingyi play, so Alvin wins
Bingyi wins 60% of the 20 games that Bingyi and Cheska play, so Bingyi
wins a total of
The games played by Cheska and Alvin do not affect Bingyi’s total
number of wins.
In total, Bingyi wins
Answer: (C)
Since
Substituting
Answer: (C)
Starting with the balls in the order
1 2 3 4 5
, we make a table of the positions of
the balls after each of the first 10 steps:
Step | Ball that moves | Order after step |
---|---|---|
1 | Rightmost | 1 2 5 3 4 |
2 | Leftmost | 2 5 1 3 4 |
3 | Rightmost | 2 5 4 1 3 |
4 | Leftmost | 5 4 2 1 3 |
5 | Rightmost | 5 4 3 2 1 |
6 | Leftmost | 4 3 5 2 1 |
7 | Rightmost | 4 3 1 5 2 |
8 | Leftmost | 3 1 4 5 2 |
9 | Rightmost | 3 1 2 4 5 |
10 | Leftmost | 1 2 3 4 5 |
After 10 steps, the balls are in the same order as at the beginning.
This means that after each successive set of 10 steps, the balls will be
returned to their original order.
Since 2020 is a multiple of 10, then after 2020 steps, the balls will be
in their original order.
Steps 2021 through 2025 will repeat the outcomes of steps 1 through 5
above, and so after 2025 steps, the balls will be in the reverse of
their original order.
Therefore, 2025 is a possible value of
Answer: (E)
The six-digit integer that Miyuki sent included the digits 2022
in that order along with two 3s.
If the two 3s were consecutive digits, there are 5 possible integers:
If the two 3s are not consecutive digits, there are 10 possible pairs of
locations for the 3s: 1st/3rd, 1st/4th, 1st/5th, 1st/6th, 2nd/4th,
2nd/5th, 2nd/6th, 3rd/5th, 3rd/6th, 4th/6th.
These give the following integers:
In total, there are thus
Answer: (E)
Solution 1
Each of the
Since there are two pieces that are each
Therefore,
The value
Note that
This also means that the other 6 pieces must also add to
We show that the value of
Since
The value
The value
The sum of the values of
(We can see that
Solution 2
The pizza is cut into 2 pieces of size
Each of these fractions can be written with a denominator of 24, so we
can think of having 2 pieces of size
To create groups of pieces of equal total size, we can now consider
combining the integers 1, 1, 2, 2, 2, 2, 3, 3, 4, and 4 into groups of
equal size. (These integers represent the size of each piece measured in
units of
Since the largest integer in the list is 4, then each group has to have
size at least 4.
Since
Here is a way of breaking the slices into
Therefore, the sum of the values of
Answer: (D)
A 10 cm by 10 cm board has 9 rows of 9 holes, or
Each hole on the 2 main diagonals has a peg in it.
There are 9 holes on each diagonal, with the centre hole on both
diagonals, since there is an odd number of holes in each row.
Therefore, the total number of holes on the two diagonals is
This means that the number of empty holes is
Answer: 64
We start by looking for patterns in the rightmost two digits of
powers of 4, powers of 5 and powers of 7.
The first few powers of 5 are
To see this, we want to understand why if the rightmost two digits of a
power of 5 are 25, then the rightmost two digits of the next power of 5
are also 25.
The rightmost two digits of a power of 5 are completely determined by
the rightmost two digits of the previous power, since in the process of
multiplication, any digits before the rightmost two digits do not affect
the rightmost two digits of the product.
This means that the rightmost two digits of every power of 5 starting
with
The first few powers of 4 are
Since 120 is a multiple of 10 and 127 is 7 more than a multiple of 10,
the rightmost two digits of
The first few powers of 7 are
Since 128 is a multiple of 4 and 131 is 3 more than a multiple of 4, the
rightmost two digits of
Therefore, the rightmost two digits of
Answer: 52
Since the shaded regions are equal in area, then when the unshaded sector in the small circle is shaded, the area of the now fully shaded sector of the larger circle must be equal to the area of the smaller circle.
The smaller circle has radius 1 and so it has area
The larger circle has radius 3 and so it has area
This means that the area of the shaded sector in the larger circle is
This means that
Thus,
Answer: 40
Since a Pretti number has 7 digits, it is of the form
From the given information, the integer with digits
Since a Pretti number is a seven-digit positive integer, then
Since
From the given information, the integer with digits
Since the thousands digit of a Pretti number is not 0, then
Since
Since the ten thousands digit and units digit of the original number are
equal, then
In other words, the units digits of
The units digit of a perfect square depends only on the units digit of
the integer being squared, since in the process of multiplication no
digit to the left of this digit affects the resulting units digit.
The squares
This gives the following table:
Units digit of |
Possible units digits of |
---|---|
0 | 0 |
1 | 1, 9 |
4 | 2, 8 |
5 | 5 |
6 | 4, 6 |
9 | 3, 7 |
Similarly, the units digit of a perfect cube depends only on the
units digit of the integer being cubed.
The cubes
This gives the following table:
Units digit of |
Possible units digits of |
---|---|
0 | 0 |
1 | 1 |
2 | 8 |
3 | 7 |
4 | 4 |
5 | 5 |
6 | 6 |
7 | 3 |
8 | 2 |
9 | 9 |
We combine this information to list the possible values of
Digit |
Possible squares | Possible cubes | Pretti numbers |
---|---|---|---|
0 | |||
1 | |||
4 | |||
5 | |||
6 | |||
9 |
For each square in the second column, each cube in the third column
of the same row is possible. (For example,
Therefore, the number of Pretti numbers is
Answer: 30
Throughout this solution, we remove the units (cm) as each length
is in these same units.
First, we calculate the distance flown by the fly, which we call
Let
Since the hexagonal base has side length 30, then
This is because a hexagon is divided into 6 equilateral triangles by its
diagonals, and so the length of the diagonal is twice the side length of
one of these triangles, which is twice the side length of the hexagon.
Also,
By the Pythagorean Theorem, since
Next, we calculate the distance crawled by the ant, which we call
Since the ant crawls
To find
Since the ant passes through
Since the ant’s path has a constant slope, its path forms the hypotenuse
of a right-angled triangle with base of length
By the Pythagorean Theorem, since
Now, we want
Since these quantities are positive, the inequality
The following inequalities are equivalent:
Answer: 19