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 110 003 is greater than 110 000 and each of the other four choices is less than 110 000, the integer 110 003 is the greatest of all of the choices.
Answer: (B)
From left to right, the number of shaded squares in each column
with shaded squares is 1, 3, 5, 4, 2.
Thus, the number of shaded squares is
Alternatively, we could note that exactly one-half of the 30 squares are
shaded since each column with shaded squares can be paired with a column
of the same number of unshaded squares. (The 1st column is paired with
the 8th, the 2nd with the 7th, the 3rd with the 6th, and the 4th with
the 5th.) Thus, again there are
Answer: (C)
Evaluating,
Answer: (C)
Since
Since
Thus,
Answer: (E)
Evaluating,
Answer: (A)
Since
Alternatively, since
Answer: (C)
Jurgen takes
Since Jurgen arrives 60 minutes before the bus leaves, he began packing
Since the bus leaves at 6:45 p.m., Jurgen began packing at 4:45 p.m.
Answer: (A)
Since the letters of RHOMBUS take up 7 of the 31 spaces on the
line, there are
Since the numbers of empty spaces on each side of RHOMBUS are the same,
there are
Therefore, the letter R is placed in space number
Answer: (B)
The digits to the right of the decimal place in the decimal
representation of
Since
This means that the 97th digit is 1, the 98th digit is 4, the 99th digit
is 2, and the 100th digit is 8.
Answer: (D)
The path that the ant walks from
The path that the ant walks from
The path that the ant walks from
This gives
Since
Thus, the total distance that the ant walks is
Answer: (D)
Suppose that the original prism has length
Since the volume of this prism is
The new prism has length
The volume of this prism, in
Answer: (E)
Since
Thus, 131 must be the integer that does not appear in Morgan’s
spreadsheet. (We note that 131 is 2 more than
Answer: (C)
The total decrease in temperature between these times is
The length of time between 3:00 p.m. one day and 2:00 a.m. the next day
is 11 hours, since it is 1 hour shorter than the length of time between
3:00 p.m. and 3:00 a.m.
Since the temperature decreased at a constant rate over this period of
time, the rate of decrease in temperature was
Answer: (B)
There are 2 possible “states” for each door: open or
closed.
Therefore, there are
If exactly 2 of the 4 doors are open, these doors could be the 1st and
2nd, or 1st and 3rd, or 1st and 4th, or 2nd and 3rd, or 2nd and 4th, or
3rd and 4th. Thus, there are 6 ways in which 2 of the 4 doors can be
open.
Since each door is randomly open or closed, then the probability that
exactly 2 doors are open is
Answer: (A)
Nasim can buy 24 cards by buying three 8-packs (
Nasim can buy 25 cards by buying five 5-packs (
Nasim can buy 26 cards by buying two 5-packs and two 8-packs (
Nasim can buy 28 cards by buying four 5-packs and one 8-pack (
Nasim can buy 29 cards by buying one 5-pack and three 8-packs (
Nasim cannot buy exactly 27 cards, because the number of cards in
8-packs that he buys would be 0, 8, 16, or 24, leaving 27, 19, 11, or 3
cards to buy in 5-packs. None of these are possible, since none of 27,
19, 11, or 3 is a multiple of 5.
Therefore, for 5 of the 6 values of
Answer: (A)
Suppose that Mathilde had
From the given information, 100 is 25% more than
From the given information, 100 is 20% less than
Therefore, at the beginning of last month, they had a total of
Answer: (E)
Suppose that
Since 68 students like lentils, these 68 students either like chickpeas
or they do not.
Since
Since 53 students like chickpeas, then
We know that there are 100 students in total and that 6 like neither
lentils nor chickpeas.
We use a Venn diagram to summarize this information:
Since there are 100 students in total, then
Therefore, there are 27 students that like both lentils and
chickpeas.
Answer: (B)
Since
Since the measures of the angles in
Answer: (D)
Before Kyne removes hair clips, Ellie has 4 red clips and
After Kyne removes the clips, the probability that Ellie chooses a red
clip is
Since Ellie starts with 4 red clips, then after Kyne removes some clips,
Ellie must have 4, 3, 2, 1, or 0 red clips.
Since the probability that Ellie chooses a red clip is larger than 0,
she cannot have 0 red clips.
Since the probability of her choosing a red clip is
Thus, the possible values of
Of these, 12 is one of the given possibilities. (One possibility is that
Kyne removes 2 of the red clips, 5 of the blue clips and 5 of the green
clips, leaving 2 red clips and 2 green clips.)
Answer: (C)
Draw one of the diagonals of the square. The diagonal passes through the centre of the square.
By symmetry, the centre of the smaller circle is the centre of the
square. (If it were not the centre of the square, then one of the four
larger circles would have to be different from the others somehow, which
is not true.)
Further, the diagonals of the square pass through the points where the
smaller circle is tangent to the larger circles. (The line segment from
each vertex of the square to the centre of the smaller circle passes
through the point of tangency. These four segments are equal in length
and meet at right angles since the diagram can be rotated by 90 degrees
without changing its appearance. Thus, each of these is half of a
diagonal.)
Since each of the larger circles has radius 5, the side length of the
square is
Since the square has side length 10, its diagonal has length
Therefore,
Of the given choices,
Answer: (C)
We follow Alicia’s algorithm carefully:
Step 1: Alicia writes down
Step 2: Since
Step 3: Alicia writes down
Step 4: Alicia sets
Step 2: Since
Step 3: Alicia writes down
Step 4: Alicia sets
Step 2: Since
Step 3: Alicia writes down
Step 4: Alicia sets
Step 2: Since
Step 3: Alicia writes down
Step 5: Since Alicia has written down five terms, she stops.
Therefore, the fifth term is 43.
Answer: 43
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
Suppose that Dewa’s four numbers are
The averages of the four possible groups of three of these are
The sums of the groups of three are equal to 3 times the averages, so
are 96, 117, 120, 132, in some order.
In other words,
Therefore,
Answer: 59
Triangular-based pyramid
Since this pyramid is built at a vertex of the cube, then
The area of
Thus, the volume of the pyramid is
Since the cube has edge length 100, its volume is
Now, 1% of 1 000 000 is
Thus, 0.01% of 1 000 000 is
This tells us that 0.04% of 1 000 000 is 400, and 0.08% of 1 000 000
is 800.
We want to determine the number of integers
This is equivalent to determining the number of integers
Since
These are the possible values for
There are
Answer: 28
Since the median of the list
Since 2023 appears more than once in the list, then it appears 5, 4, 3,
or 2 times.
Case 1: 2023 appears 5 times.
Here, the list is 2023, 2023, 2023, 2023, 2023.
There is 1 such list.
Case 2: 2023 appears 4 times.
Here, the list would be 2023, 2023, 2023, 2023,
Since the mean of the list is 2023, the sum of the numbers in the list
is
There are 0 lists in this case.
Case 3: 2023 appears 3 times.
Here, the list is
In the first case, the mean of the list is less than 2023, since the sum
of the numbers will be less than
In the third case, the mean of the list is greater than 2023, since the
sum of the numbers will be greater than
So we need to consider the list
Since the mean of this list is 2023, then the sum of the five numbers is
Since
Since there are 2022 choices for
Case 4A: 2023 appears 2 times;
(We note that if 2023 appears 2 times, then since
Here, the list is
This list has median 2023 and no other integer appears more than once.
Thus, it still needs to satisfy the condition about the mean.
For this to be the case, the sum of its numbers equals
Every pair of values for
If
If
Each time we increase
Therefore, the number of pairs of values for
Case 4B: 2023 appears 2 times;
Here, the list is
This list has median 2023 and no other integer appears more than once.
Thus, it still needs to satisfy the condition about the mean.
For this to be the case, the sum of its numbers equals
If
If
As
Therefore, the number of pairs of values for
Combining all of the cases, the total number of lists
The sum of the digits of
Answer: 28