Wednesday, February 22, 2023
(in North America and South America)
Thursday, February 23, 2023
(outside of North American and South America)
©2022 University of Waterloo
Since each of the three fractions is equal to 1, then
Answer: (C)
Since
Alternatively, we could note that
Answer: (D)
We add 25 minutes to 1 hour and 48 minutes in two steps.
First, we add 12 minutes to 1 hour and 48 minutes to get 2 hours.
Then we add
Alternatively, we could note that 1 hour and 48 minutes is
Answer: (A)
On Day 1, Lucy sees 2 blue jays and 3 cardinals, and so sees 1
more cardinal than blue jay.
On Day 2, Lucy sees 3 blue jays and 3 cardinals.
On Day 3, Lucy sees 2 blue jays and 4 cardinals, and so sees 2 more
cardinals than blue jays.
Thus, over the three days, Lucy saw
Answer: (B)
When looking at through a window from behind, the digits appear as mirror images
and in reverse order, and so appear as
.
Answer: (C)
Solution 1
Since
Since the measures of the angles in
Solution 2
Since the measures of the angles in
Answer: (E)
A cube has six identical faces.
If the surface area of a cube is 24, the area of each face is
Since each face of this cube is a square with area 4, the edge length of
the cube is
Thus, the volume of the cube is
Answer: (E)
If
Since there are 4 green beads, the total number of beads must be
Thus, Charlie needs to add
Answer: (A)
Solution 1
Suppose that the original number is 100.
When 100 is increased by 60%, the result is 160.
To return to the original value of 100, 160 must be decreased by
60.
This percentage is
Solution 2
Suppose that the original number is
When
To return to the original value of
This percentage is
Answer: (E)
Since each door can be open or closed, there are 2 possible
states for each door.
Since there are 5 doors, there are
If the doors are labelled P, Q, R, S, T, the pairs of doors that can be
opened are PQ, PR, PS, PT, QR, QS, QT, RS, RT, ST. There are 10 such
pairs.
Therefore, if one of the 32 combinations of states is chosen at random,
the probability that exactly two doors are open is
Answer: (A)
After Karim eats
Since he divides these candies equally among his three children, the
integer
If
If
Therefore,
Answer: (C)
A rectangle that is 6 m by 8 m has perimeter
If posts are put in every 2 m around the perimeter starting at a corner,
then we would guess that it will take
The diagram below confirms this:
Answer: (B)
Since
Furthermore, the exponents of the prime factors of a perfect square must
be all even.
Therefore, any perfect square that is a multiple of 2023 must be
divisible by
Therefore, the smallest perfect square that is a multiple of 2023 is
We can check that
Answer: (C)
Since
Since
Since
Thus,
Answer: (E)
Since
Thus,
Since
Thus,
Answer: (B)
Solution 1
Since
We know that
Since
Suppose that
Then
The height of the trapezoid is the length of
Since the area of the trapezoid is
Solution 2
Let
Rectangle
The area of
The area of
Answer: (A)
Megan’s car travels 100 m at
Hana’s car completes the 100 m in 5 s fewer, and so takes 75 s.
Thus, the average speed of Hana’s car was
Answer: (C)
The number of bars taken from the boxes is
If these bars all had mass 100 g, their total mass would be 3100
g.
Since their total mass is 2920 g, they are
Since all of the bars have a mass of 100 g or of 90 g, then it must be
the case that 18 of the bars are each 10 g lighter (that is, have a mass
of 90 g).
Thus, we want to write 18 as the sum of two of 1, 2, 4, 8, 16 in order
to determine the boxes from which the 90 g bars were taken.
We note that
Answer: (B)
Since the average of
Since the average of
Since the average of
Thus,
We note that
Answer: (D)
Each group of four jumps takes the grasshopper 1 cm to the east
and 3 cm to the west, which is a net movement of 2 cm to the west, and 2
cm to the north and 4 cm to the south, which is a net movement of 2 cm
to the south.
In other words, we can consider each group of four jumps, starting with
the first, as resulting in a net movement of 2 cm to the west and 2 cm
to the south.
We note that
Thus, after 79 groups of four jumps, the grasshopper is
The grasshopper has made
After the 317th jump (1 cm to the east), the grasshopper is 157 cm west
and 158 cm south of its original position.
After the 318th jump (2 cm to the north), the grasshopper is 157 cm west
and 156 cm south of its original position.
After the 319th jump (3 cm to the west), the grasshopper is 160 cm west
and 156 cm south of its original position.
After the 320th jump (4 cm to the south), the grasshopper is 160 cm west
and 160 cm south of its original position.
After the 321st jump (1 cm to the east), the grasshopper is 159 cm west
and 160 cm south of its original position.
After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west
and 158 cm south of its original position.
After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west
and 158 cm south of its original position, which is the desired
position.
As the grasshopper continues jumping, each of its positions will always
be at least 160 cm south of its original position, so this is the only
time that it is at this position.
Therefore,
Answer: (A)
Since the line with equation
Since
Therefore, the possible values of
The sum of the possible values of
Answer: 93
From the given information, if
Since all of the numbers that we can use are positive, then
This means that the largest integer in the list, which is 13, cannot be
either
Thus, for
Therefore, the next largest possible value for
Here, we could have
The remaining integers (4, 5 and 6) can be put in the shapes in the following way that satisfies the requirements.
This tells us that the largest possible value of
Answer: 20
Since
Since
Thus,
We add some additional labels to the diagram:
We note that the angles between the straight lines at
Since
Since the side length of each square is 10, then
Since
Since
Since
Thus,
Since
Thus,
Since
Using similar reasoning,
The total area,
Therefore,
Answer: 61
We want to determine the probability that Carina wins 3 games
before she loses 2 games.
This means that she either wins 3 and loses 0, or wins 3 and loses
1.
If Carina wins her first three games, we do not need to consider the
case of Carina losing her fourth game, because we can stop after she
wins 3 games.
Putting this another way, once Carina has won her third game, the
outcomes of any later games do not affect the probability because wins
or losses at that stage will not affect the question that is being
asked.
Using W to represent a win and L to represent a loss, the possible
sequence of wins and losses that we need to examine are WWW, LWWW, WLWW,
and WWLW.
In the case of WWW, the probabilities of the specific outcome in each
of the three games are
Therefore, the probability of WWW is
In the case of LWWW, the probabilities of the specific outcome in
each of the four games are
Therefore, the probability of LWWW is
Using similar arguments, the probability of WLWW is
Here, we used the fact that the probability of a loss after a win is
Finally, the probability of WWLW is
Therefore, the probability that Carina wins 3 games before she loses
2 games is
The sum of the numerator and denominator of this fraction is 23.
Answer: 23
Solution 1
Let
We note that
Therefore,
We want to write
There are several situations to consider.
First, we look at possible sets
Second, we look at possible sets
Third, we rule out possible sets
Fourth, we rule out possible sets
Case 1:
Here,
If
These give the following corresponding values of
Note that
If
If
Therefore, in this case, there are 15 possible sets.
Case 2:
Here, we have
Note that
Suppose that
If
If
Suppose that
If
Since
Therefore, in this case, there are 3 possible sets.
Case 3:
Suppose that
If
We cannot have
Suppose that
If
As in Case 2, we cannot have
Therefore, in this case, there are 3 possible sets.
Case 4:
Suppose that
If
We cannot have
If
As in Cases 2 and 3, we cannot have
Therefore, in this case, there are 3 possible sets.
Case 5:
Here,
Since
Since there do not exist two distinct odd integers greater than 1 with a
product less than 15, there are no possible sets in this case.
A similarly argument rules out the products
Case 6:
Note that
Here,
Since
Having considered all cases, there are
Solution 2
We note first that
Therefore,
Since
Since
Using this information, we can eliminate many possibilities for
Thinking about each of these as
To write
This leaves the integers 21, 25, 27, 35, 39, 45, 49, 63, 65, 75, 81,
91.
Seven of these remaining integers are equal to the product of two
prime numbers, which are either equal primes or at least one of which is
equal to 7, 11 or 13. These products are
In each case,
Similarly,
This leaves the integers 27, 45, 63, 75, 81 to consider.
Consider
Consider
Since there cannot be two 3s or two 9s in the set and there must be two
powers of 3 in the set, there are four possibilities for the set
Since two of these prime factors are 3, they cannot each be an
individual element of the set and so one of the 3s must always be
combined with another prime giving the following possibilities:
Using a similar argument to that in the case of
We have determined that the total number of sets is thus
Answer: 24