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Thursday, April 4, 2024
(outside of North American and South America)
©2024 University of Waterloo
For every
Therefore, when
Alternatively, when
By the Pythagorean Theorem in
Alternatively, we could remember the Pythagorean triple
Since
Therefore,
Suppose that the sequence is
Since
Since
Since
Since
Therefore, the first term is
From the given information, we obtain
Since
Suppose that Jimmy’s marks on his first, second, third, and
fourth tests were
Since Jimmy’s average on his first, second and third tests was
Since Jimmy’s average on his second, third and fourth tests was
Since his mark on the fourth test was
Thus,
Subtracting the first equation from the second equation, we obtain
Since
Thus,
With
With
From the given information
Therefore,
Alternatively, we could note that the difference in total prices is the
difference in the amount of tax paid. This is the difference between
When
Since
When
Since
When
Since
Therefore,
Since
Similarly,
Further,
Since
Since
Since
Thus,
Alternatively, if
Thus, the area of
The line with equation
To find the
Since
Since the area of
Next, we note that the line with equation
This means that the area of
We can determine the area of
Now, we can consider
Thus,
This means that
Since
Therefore, the coordinates of
Suppose that
We use the notation
Thus,
Then,
If
If
Since we are told that there are three possible values for
(Can you explain why there are exactly three such values?)
The
Therefore, the discriminant is maximized when
Every multiple of
For such a multiple to be between 10 000 and 100 000, we need
Since
We also want the units digit of
This means that the units digit of
Therefore, the possible values of
There are
Thus,
Solution 1:
We can partition the
In Math Club | Not in Math Club | |
---|---|---|
In Physics Club | ||
Not in Physics Club |
From the given information, there are
Among the students in the physics club, twice as many are not in the
math club as are in the math club. This means that
From the given information, there are
Since
Since
Lastly, we know that
Since each of
Therefore,
Solution 2:
Since there are
Since each of
Since
Thus, we let
In this case,
Now, among the
In other words,
Therefore,
Since
Thus, the number of students in the physics club is
Also, the number of students in the math club is
Finally, we know that
there are
Therefore, the number of students in neither club is
Suppose that the length of the track is
When Arun and Bella run over the same interval of time, the ratio of the
distances that they run is equal to the ratio of their speeds.
Consider the interval of time from the start to when they first meet. In
the diagram,
Since Arun has run
Thus,
From their first meeting point
Over this time, Arun runs from
Since Bella runs
Thus, over this second interval of time,
Equating expressions for
Checking, if the length of the track is
This means that from the start to
Also, from
Note that
Using exponent laws, the following equations are equivalent:
Since
Listing these in increasing order, the solutions to the original
equation are
We join
Since
Since the radius of the circle with centre
Since the radius of the circle with centre
Let
We note that
Since
By the Pythagorean Theorem,
Since
Finally, the area of
If we joined
Since
First, we note that
We also note from the original system of equations that
Therefore, we can re-write the original system of equations as
Thus,
Since
Similarly,
Therefore,
We can check by substitution that this triple does satisfy the original
system of equations.
Suppose that a sequence of
This means that
Since
Thus, we need to count the number of sequences of
This is the same as creating a sequence of
There are
(Alternatively, we could see that there are
Solution 1:
Since
This means that we can think about sequences of
In this context, a direction change happens when the sequence changes
from a block of one letter to a block of another letter (that is, when
the sequence has an occurrence of "RL" or "LR").
For the sequence to change directions an even number of times, the
number of blocks of letters in the sequence is odd.
For the number of blocks to be odd, the sequence must begin and end with
the same letter.
If the sequence of
If the sequence of
In total, there are
Solution 2:
As in Solution 1, we want to count the number of sequences of
Since there are
Case 1: There are
Suppose that the block order is R-L-R.
Here, there are
There are
This means that there are
Suppose that the block order is L-R-L.
The block of R’s includes
There are
This means that there are
In total, there are
Case 2: There are
Since there cannot be
Each block of L’s includes exactly
There are
Therefore, there are
Case 3: There are
Suppose that the block order is R-L-R-L-R.
The two blocks of L’s include
The three blocks of L’s include
Starting with one R in each block, there are
There are
There are
There is
Thus, there are
Suppose that the block order is L-R-L-R-L.
Each block of L’s includes exactly
There are
This means that there are
In total, there are
Combining the cases, there are
Consider the sequences of length
Suppose that such a sequence includes
Since
This means that all sequences of length
Before proceeding to the general case, we deal with some specific small
values.
When
When
When
When
Therefore, we assume that
We think about sequences of
As in (b) Solution 1, the number of direction changes is even exactly
when the number of blocks of letters is odd, which is exactly when the
sequence begins and ends with the same letter.
For such a sequence to begin and end with R, the
For such a sequence to begin and end with L, the
Thus, there are
This is true for exactly half of all sequences when the following
equivalent equations are true:
To finish the problem, we now need to count the number of perfect
squares (that is, possible values for
Since
Here is the general set-up for this problem along with the
specific instance in (a) where
Since
Since
Alternatively, we could note that the area of
If
Since
Since
Thus, the coordinates of
We note that the area of
Suppose that
This is true exactly when
the area of
the area of
the area of
(Bullets 1 and 2 tell us that the area of quadrilateral
Bullets 1 and 3 tell us that the area of quadrilateral
Since three of the areas are
The line through
The line through
Since
Therefore,
Since
Therefore,
Next, we find the coordinates of
The slope of the line through
The slope of the line through
To find the
We look for pairs of the form
From the relationship from (b),
Since
We need to have
Finally, we need
Thus,
Since
Therefore, when
We note that there are infinitely many rational numbers in any interval
of non-zero length; in particular, there are infinitely many rational
numbers
We do need to confirm that the values of
To do this, we show that if
When
This means that if
Therefore, the pairs