June 2022
©2022 University of Waterloo
Question | Answer |
---|---|
1 | 3.001 |
2 | 2 |
3 | 8 |
4 | B, C, D |
5 | 2 |
6 | 4 |
7 | 64 |
8 | 108 |
9 | 2 |
10 | 2 |
11 | rotate |
12 | 3 |
13 | 60 |
14 | 180 |
15 | 7 |
5 | B | 2 | 7 | 7 | B | 1 | 0 | 8 |
8 | 5 | 7 | B | 9 | 7 | 8 | B | 2 |
5 | B | B | 3 | B | 5 | B | 3 | 6 |
B | 1 | 9 | 9 | 1 | B | 9 | 9 | 0 |
4 | 5 | B | 5 | B | 3 | B | 2 | 4 |
5 | 8 | 7 | B | 5 | 3 | 4 | 6 | B |
6 | 0 | B | 2 | B | 6 | B | B | 1 |
7 | B | 1 | 9 | 6 | B | 6 | 1 | 6 |
8 | 8 | 1 | B | 4 | 6 | 6 | B | 2 |
Day of the Week | ||||||
---|---|---|---|---|---|---|
Monday | Tuesday | Wednesday | Thursday | Friday | ||
Kind of Animal | cows | Corina | Aditi | Corina | Brigid | Aditi |
goats | Aditi | Corina | Eliel | Corina | Corina | |
pigs | Eliel | Damien | Aditi | Damien | Damien | |
sheep | Damien | Eliel | Damien | Aditi | Brigid | |
horses | Brigid | Brigid | Brigid | Eliel | Eliel |
Player | Player 1 | Player 2 | Player 3 | Player 4 |
---|---|---|---|---|
Correct Answer | 15 | 25 | 5 | 3 |
Player | Player 1 | Player 2 | Player 3 | Player 4 |
---|---|---|---|---|
Correct Answer | 4 | 32 | 337 | 1209 |
Player | Player 1 | Player 2 | Player 3 | Player 4 |
---|---|---|---|---|
Correct Answer | 24 | 3 | 28 | 7 |
Player | Player 1 | Player 2 | Player 3 | Player 4 |
---|---|---|---|---|
Correct Answer | 4 | 24 | 744 | 1765 |
When we arrange the five numbers from least to greatest, we
obtain
Answer:
A square has exactly four lines of symmetry: one vertical, one horizontal, and two diagonal, as shown in the following diagrams.
We can divide each of these diagrams into nine smaller squares and shade the three in the top row and the three in the bottom row, as shown.
The vertical and horizontal lines still divide the diagram into two
halves so that one half is the reflection of the other.
The two diagonal lines divide the diagram into two halves, but the two
halves are not reflections of one another any longer.
Thus, the diagram has a vertical line of symmetry and a horizontal line
of symmetry, for a total of
Answer:
To determine the input, we work backwards through the
flowchart.
If the output is
Then the number before we subtract
Finally, the number before we multiply by
Answer:
We can show that the triangle with vertices
The base
For completeness, we show that the other triangles do not have an
area of
In
In
To find the area of
Then the area of
The area of rectangle
The area of
The area of
The area of
Thus, the area of
Answer:
Reading from the bar graph, there were
Since half the hats sold were size L, the total number of hats sold was
From the graph, we can see that
Thus, a total of
This means that
Answer:
The three flowering plants can end up next to each other by
first swapping the leftmost flowering plant once towards the right, and then
swapping the rightmost flowering plant three times towards the left.
Every non-flowering plant must be involved in a swap because each
non-flowering plant is positioned between two flowering plants. Swapping
two non-flowering plants does not have any effect so each useful swap of
a non-flowering plant must be with a flowering plant.
There are four non-flowering plants and so there must be at least four
swaps.
Notice that moving the flowering plants all the way left or all the way
right requires more than four swaps.
Answer:
The even integers between
Next, we want to find a number
We notice that our lower bound,
By increasing each of the
These odd integers are
That is, the number of even integers between
Answer:
First, Chance collects
On his
In total, Chance collects
Answer:
The first row is missing a
To complete the upper left
The second row is now missing a
To complete the fourth column, we place a
Now the
The completed grid is shown.
Answer:
The second scale shows that has the same mass as
.
If we double what is on both sides of this scale, then has the same mass as
.
From the first scale, also has the same mass as
.
Therefore, must have the same mass as
, or
must have the same mass as
.
Since has the same mass as
and
has the same mass as
it follows that
has the same mass as
, as shown.
The third scale shows that also has the same mass as
.
So must have the same mass as
.
If has a mass of
must have a mass of
Since each triangle has the same mass, has a mass of
Answer:
In total, six actions are performed. Five of the actions are described and one is unknown as a result of the missing block of code. After each action, the face’s orientation changes.
In the following diagram, seven squares are arranged from left to right, which will represent the seven orientations the face takes as the code is executed. Since the face starts and ends in the same orientation, we draw the same face in the first and last squares. In between two side-by-side squares is an arrow pointing right. Over top of the arrows we write the block names (using FV, FH, and R for short) in the order that the code executes them. The question mark over top of the fourth arrow indicates the missing block of code.
Following the code from the beginning, we can determine the orientation of the face after the first three actions have been performed.
After the missing block, two more actions are performed and the final orientation of the face is the same that it started in. Working backwards, we can determine the orientation of the face in the remaining two positions.
Examining the face in the fourth and fifth position, we can conclude that the missing block should be rotate.
Answer: rotate
The volume of the cheese before Raissa’s cut is
The volume of the remaining cheese is
Since
Answer:
Since the digit
For each of these six configurations, there are two ways for the
digits
Therefore, there are
At the very most, Anil will have to try all
Answer:
If Mirela and Kumara have sold
Since
But,
Since
Answer:
Since there are eight given integers, the smallest nada set could contain one integer and the largest nada set could contain eight integers.
Case 1: Sets containing one integer
All of the given integers are non-zero so Elise cannot make any nada sets containing exactly one integer.
Case 2: Sets containing two integers
The given integers do not contain any pairs that add to zero, so Elise cannot make any nada sets containing two integers.
Case 3: Sets containing three integers
If three integers have a sum of zero then either one integer is positive or two integers are positive. Can you verify this for yourself?
What if one integer is positive?
If one integer is positive then the other two must be negative. The
possible sums of the two negative integers are:
There are no nada sets containing three integers where one is
positive.
What if two integers are positive?
If two integers are positive then the third integer must be negative.
The possible sums of the two positive integers are:
Thus,
Elise can make two different nada sets containing three integers.
Case 4: Sets containing four integers
If a set of four integers has a sum of zero then either one integer is positive, two integers are positive, or three integers are positive.
What if one integer is positive?
If one integer is positive then the other three integers must be
negative.
The positive integers are between
The possible sum of three negative integers is between
Thus, any sum of one positive integer and three negative integers will
always be less than zero, so no such nada sets can be found.
What if two integers are positive?
If a set of two positive and two negative integers has a sum of zero,
then the sum of the positive integers must be opposite to the sum of the
negative integers.
Using the sums found in Case 3, we have that
What if three integers are positive?
If three integers are positive then the fourth integer must be
negative.
The possible sum of three positive integers is between
The negative integers are between
Thus, any sum of three positive integers and one negative integer will
always be greater than zero, so no such nada sets can be found.
Elise can make two different nada sets containing four integers.
Case 5: Sets containing five integers
The given eight integers have a total sum of zero. Thus, if a set of
five integers has a sum of zero, then the remaining three integers must
also have a sum of zero.
Can you verify this for yourself?
From before, we know that there are two different sets of three integers
that have a sum of zero, namely
Thus, there are also two different sets of five integers that have a sum
of zero, namely
Elise can make two different nada sets containing five integers.
Case 6: Sets containing six integers
Similarly, since the sum of a set containing any two of the given
integers cannot equal zero, it follows that the sum of a set containing
any six of the given integers cannot equal zero.
Elise cannot make any nada sets containing six integers.
Case 7: Sets containing seven integers
Since the sum of a set containing any one of the given integers
cannot equal zero, it follows that the sum of a set containing any seven
of the given integers cannot equal zero.
Elise cannot make any nada sets containing seven integers.
Case 8: Sets containing eight integers
The sum of all eight integers equals zero, so Elise can make one nada set containing eight integers.
Therefore, Elise can make
Answer:
5 | B | 2 | 7 | 7 | B | 1 | 0 | 8 |
8 | 5 | 7 | B | 9 | 7 | 8 | B | 2 |
5 | B | B | 3 | B | 5 | B | 3 | 6 |
B | 1 | 9 | 9 | 1 | B | 9 | 9 | 0 |
4 | 5 | B | 5 | B | 3 | B | 2 | 4 |
5 | 8 | 7 | B | 5 | 3 | 4 | 6 | B |
6 | 0 | B | 2 | B | 6 | B | B | 1 |
7 | B | 1 | 9 | 6 | B | 6 | 1 | 6 |
8 | 8 | 1 | B | 4 | 6 | 6 | B | 2 |
From the grid, the hundreds digit of this number is
From the grid, the hundreds digit of this number is
The positive difference is
The median of any three digits is an integer, and so
the mean must also be an integer.
From the grid, the hundreds digit of this number is
If the tens digit is
The last two digits of
From the grid, the thousands digit of this number is
From the grid, the tens digit is 9. It follows that the hundreds digit
is 9.
Thus, the number must be
From the grid, the tens digit of this number is
From the grid, the ones digit of this number is
One square has
The number of nickels is
The volume is equal to
Since
From the grid, the hundreds digit of this number is
The sum of the digits in
From the grid, the hundreds digit of this number is
From the grid, the tens digit of this number is
From the grid, the ones digit of this number is
Two-fifths of
From the grid, the digits
From the grid, the tens digit of this number is
The product is
In
From the grid, the thousands digit of this number is
In
The smallest prime number greater than
Since
The only factors of
From the grid, the tens digit of this number is
A triangle with area
Thus, the number is 66.
We start by considering clues (2) and (5):
Damien feeds the sheep on only Monday and Wednesday.
Damien feeds only the pigs and the sheep.
Since Damien feeds the sheep on only Monday and Wednesday, and Damien feeds only the pigs and the sheep, it follows that Damien must feed the pigs on Tuesday, Thursday, and Friday.
Next we consider clue (6):
Every day that Damien feeds the pigs, Corina feeds the goats.
Since Damien feeds the pigs on Tuesday, Thursday, and Friday, it follows that Corina feeds the goats on Tuesday, Thursday, and Friday.
Next we consider clue (8):
The student who feeds the cows on Monday also feeds the goats on Tuesday.
Since Corina feeds the goats on Tuesday, it follows that Corina must feed the cows on Monday.
The following partially-completed table contains the information we have determined so far.
Day of the Week | ||||||
---|---|---|---|---|---|---|
Monday | Tuesday | Wednesday | Thursday | Friday | ||
Kind of Animal | cows | Corina | ||||
goats | Corina | Corina | Corina | |||
pigs | Damien | Damien | Damien | |||
sheep | Damien | Damien | ||||
horses |
Next we consider clues (3), (1), (4), and (10):
The same student feeds the horses on Monday, Tuesday, and Wednesday.
The student who feeds the cows on Thursday also feeds the horses on Wednesday.
Eliel feeds all the animals except for the cows.
Aditi does not feed the same kind of animal more than twice each week.
Clues (3) and (1) tell us that one particular student feeds the horses on Monday, Tuesday, and Wednesday, and the cows on Thursday. Clues (4) and (10) tell us that this student is not Eliel or Aditi. On Monday Corina and Damien are already feeding other animals, so the only student left is Brigid. Thus, Brigid must feed the horses on Monday, Tuesday, and Wednesday, and the cows on Thursday.
Now on Tuesday, the only remaining students are Aditi and Eliel, and the only remaining animals are the cows and the sheep. From clue (4) we can determine that on Tuesday, Aditi feeds the cows and Eliel feeds the sheep.
Next, we consider clue (9):
The student who feeds the goats on Monday also feeds the cows on Tuesday.
Since Aditi feeds the cows on Tuesday, it follows that Aditi feeds the goats on Monday. Then Eliel is the only student left on Monday, so Eliel must feed the pigs on Monday.
The following partially-completed table contains the information we have determined so far.
Day of the Week | ||||||
---|---|---|---|---|---|---|
Monday | Tuesday | Wednesday | Thursday | Friday | ||
Kind of Animal | cows | Corina | Aditi | Brigid | ||
goats | Aditi | Corina | Corina | Corina | ||
pigs | Eliel | Damien | Damien | Damien | ||
sheep | Damien | Eliel | Damien | |||
horses | Brigid | Brigid | Brigid |
Next we look back at clue (4):
Eliel feeds all the animals except for the cows.
This clue tells us that Eliel feeds the goats at least one day of the week. Since Wednesday is the only day left for the goats to be fed, it follows that Eliel must feed the goats on Wednesday.
Next we look at clue (7):
The sheep are fed by four different students each week.
The sheep are being fed by Damien and Eliel on Monday, Tuesday, and Wednesday. Thus, on Thursday and Friday the sheep must be fed by two different students, who are not Damien or Eliel. On Thursday, the only available students are Aditi and Eliel, and so Aditi must feed the sheep leaving Eliel to feed the horses. On Friday, the only available students are Aditi, Brigid, and Eliel. The only one of those who has not already been assigned to the sheep is Brigid, so Brigid must feed the sheep on Friday.
From clue (4) we know that Eliel does not feed the cows, so on Friday, Eliel must feed the horses and Aditi must feed the cows.
Finally, we look at clue (10):
Aditi does not feed the same kind of animal more than twice each week.
On Wednesday, the only available students are Aditi and Corina, and the only available animals are the pigs and the cows. Since Aditi feeds the cows on Tuesday and Friday, it follows that he cannot also feed them on Wednesday. So on Wednesday Aditi must feed the pigs and Corina must feed the cows.
This completes the logic puzzle.
Day of the Week | ||||||
---|---|---|---|---|---|---|
Monday | Tuesday | Wednesday | Thursday | Friday | ||
Kind of Animal | cows | Corina | Aditi | Corina | Brigid | Aditi |
goats | Aditi | Corina | Eliel | Corina | Corina | |
pigs | Eliel | Damien | Aditi | Damien | Damien | |
sheep | Damien | Eliel | Damien | Aditi | Brigid | |
horses | Brigid | Brigid | Brigid | Eliel | Eliel |
(Note: Where possible, the solutions are written as if the value of
P1: Evaluating,
P2: The fruits that are not apples are the oranges,
pears, and bananas. The total number of these is
Since the answer to the previous question is
P3: The numbers
So if
If
Since the answer to the previous question is
P4: After the elevator went up
Answer:
P1: The number line shown has length
The number line is divided into three equal parts, and so each part has
length
P2: After Josh presses the button once, there will be
After he presses the button twice, there will be
After he presses the button three times, there will be
Since the answer to the previous question is
P3: There are
The sum of the numbers in the list is
The first
The
Since the answer to the previous question is
P4: The smallest positive integer greater than
The sum of these three values is
Since the answer to the previous question is
Answer:
P1: Since
Then,
Then,
P2: In total, Stefanie gave the group
Since the answer to the previous question is
Since
P3: The width of the rectangle is
Since the answer to the previous question is
P4: Since there are
Since the answer to the previous question is
Answer:
P1: If we choose the coins with the largest possible values
first, then we will use the fewest number of coins. This can be done
using two quarters, one dime, and one nickel.
The total value of these coins is
There are
P2: The distance between the houses is
If Vivek bikes there and back, then in total, Vivek bikes
Since the answer to the previous question is
P3: Since Anar plants
After
P4: We can convert each of the units so that each mass is written
in grams.
If
Since the answer to the previous question is
Answer: