Wednesday, November 13, 2024
(in North America and South America)
Thursday, November 14, 2024
(outside of North American and South America)
©2024 University of Waterloo
Solution 1
The container is in the shape of a cube, so no matter which face is
sitting on the table, the base of the container is a square with side
length
The space occupied by the water in the container is a rectangular prism.
The base of this rectangular prism is the base of the container and its
height is the depth of the water.
The volume of the water is
Solution 2
The container is in the shape of a cube, so no matter which face is
sitting on the table, the base of the container is a square with side
length
The depth of the water is half the height of the cube, so the volume of
the water is half the volume of the cube.
The volume of the cube is
Answer:
Solution 1
Suppose the number of students surveyed is
From the pie graph, the number of students who chose green is
All other students chose red, so the number of students who chose red is
From the bar graph, the number of students who chose red is
Dividing both sides of this equation by
Solution 2
From the pie graph,
The rest of the students chose red, so the percentage of the students
that chose red must be
This means
From the bar graph,
Therefore, the
Answer:
In an equilateral triangle, each angle measures
In a square, each angle measures
Using this, we have
Since
Since
The angles in a triangle have a sum of
Substituting
Using again that
Answer:
Suppose that
Then
As well,
We have shown that when
Now suppose
If we insist that
Since
Thus, if we assume
Putting
It is given that
Therefore, the greatest possible value of
Note: To see how one might arrive at the number
Multiplying by
In the inequality
Add
Combining
The answer of
Answer:
We will refer to the seven columns from left to right as C1, C2,
and so on to C7.
In every row, the integer in C1 is
In the first row, the integer in C2 is
The first two integers in the
The third integer in the
The fourth integer in the
The fifth integer in the
The sixth integer in the
The seventh integer in the
To summarize, the integers in the
In C2 through C7, the integers increase moving down the column, so none
of these columns contains the same integer multiple times.
We conclude that we are looking for two-digit integers appearing in
exactly five of the columns.
The integers in C2 are
The integers in C3 are
Thus, C2 and C3 each contain every two-digit
integer, so we are really looking for two-digit integers that are in
exactly
The integers in C7 are those of the form
All other integers in C7 have at least three digits, so the only
two-digit integers in C7 are
We will consider two cases: Integers not in C7 but in C4, C5, and C6,
and integers in C7 and in exactly two of C4, C5, and C6.
Case 1: Two-digit integers that are in C4, C5, and C6, but not in C7.
Suppose a two-digit integer
The integers in C6 are of the form
The integers in C4 are of the form
We conclude that
We need to check which of these values of
In general, to determine whether an integer
For example,
Using the idea in the previous paragraph, we will solve
The
Only
Neither
Case 2: Two-digit integers that are in C7 and exactly two of C4, C5, and C6.
In the table below, we check each of the two-digit integers in C7 to see in which of C4, C5, and C6 they also appear.
solution to |
in C4? | solution to |
in C5? | solution to |
in C6? | # of columns | |
---|---|---|---|---|---|---|---|
YES | NO | YES | 2 | ||||
YES | YES | NO | 2 | ||||
YES | NO | NO | 1 | ||||
YES | NO | NO | 1 | ||||
YES | YES | NO | 2 |
From the table, we see that
Hence, of the two-digit integers in C7, exactly three are in exactly
five cells in the grid.
Combining the two cases, there are five two-digit integers that are in
exactly five cells. They are
Answer:
Solution 1
We will compute the probability that the condition fails, then
subtract the result from
The given condition is that at least one diameter has a multiple of
For this condition to fail, there must be no diameter with this
property.
There are only two available multiples of
Therefore, to ensure the condition fails, we need to ensure that the
diameters containing
Consider the diameter with
Similarly, the diameter with
Therefore, the condition fails exactly when
There are
Once
Continuing in this way, once
Because of this, there are
Next, we will count the ways to place the integers so that
There are
After the integer
Once these integers are placed, there are
One of the integers from
Thus, there are
The remaining four integers, whatever they happen to be, can now be
placed arbitrarily and the condition will still be guaranteed to
fail.
By reasoning similar to earlier, there are
The probability that the condition fails is
Solution 2
As explained in Solution 1, there are
The only multiples of
We will consider four cases.
Case 1:
In this case,
There are
There are
Therefore, there are
In the remaining three cases,
Case 2:
There are
The remaining even integers are
After these four integers are placed, there are four remaining integers
that can be placed in any of
There are
Case 3:
As with the previous case, there are
The remaining even integers are
After these integers are placed, there are
There are
Case 4:
This case has the same count as the previous case.
The probability is
Answer:
Throughout this solution, when we refer to the "difference"
between two numbers, we mean the larger of the two numbers minus the
smaller. For example, the difference between
The coordinates of
The length of
The length of
The area of
The height from
The length of
The area of
This area is given to be
Quadrilateral
The height of
The height of
The length of
The area of
The area of
The area of quadrilateral
Since
Two such integers are
To see how to arrive at
The time that it takes to run
The total time it took Beryl to run
The time that it took Carol to walk the first
The time that it took Carol to walk the remaining
In total, the walk took
We now have the equation
Multiplying through by
Rearranging this equation gives
The time that Daryl was riding at
The time that Daryl was riding at
The total time that Daryl was riding was
Multiplying through by
Since
In order for
Substituting
Since
Substituting
Since
There are
By similar reasoning to that which was used at the beginning of
the solution to part (c), Errol spent
Thus,
We are looking for positive integer solutions to this equation.
Since
There is some positive integer
Dividing through by
Since
Substituting this into
Since every value of
Suppose
There are
If
Continuing in this way, there are
When
Therefore, the number of positive integer solutions is
The 3-sign sum is
Solution 1
Consider a slice of the list
From the previous paragraph, we deduce that the slices of
The quantity
In other words,
Using this, we get that the quantity
For example,
The way the sums are expressed above, it might become apparent that
there is simpler way to see that this difference equals
To generalize this, we will introduce the notation
With this notation and from the expression above, we can see that
These 3-sign sums are identical
except for the appearance of
Thus,
More generally, if
Specifically,
We can use this to find the exact value of the expression
The quantity
The quantity
Continuing, we have that
This pattern continues and we get that
Therefore, we have
Solution 2
With notation and reasoning as in Solution 1, we have that
For example,
It occurs in
The integer
The integer
Continuing in this way,
In general, if
Suppose
The integer
Continuing, it occurs in
This pattern continues with two
If
Observe that the contribution never depends on
Thus, in the quantity
From the calculations above, since
Solution 1
Using the work from part (b),
First, let’s focus on the sum of the first nine terms
The 3-sign sums
By similar reasoning with
Continuing in this way, we can total all occurrences of
In conclusion, in the sum
Thus,
Therefore,
By nearly identical reasoning, the sum of the next nine terms,
The pattern of
As well, the integers in the second triangular sum can be obtained by
adding
There are
Hence, the sum in (***) is
Thus, the sum
The same will be true for the sum of the next nine terms in
Since
We conclude that
Solution 2
In this solution, we will use the notation and some calculations from
the solutions to part (b). Specifically, we will explicitly compute
From Solution 2 to part (b), if
The quantity
Every integer is either a multiple of
The multiples of
Observe that
The integers that are