Tuesday, April 7, 2020
(in North America and South America)
Wednesday, April 8, 2020
(outside of North American and South America)
©2020 University of Waterloo
Solution 1:
If
In other words, for every
Therefore, when
Solution 2:
When
Solution 1:
The point at which a line crosses the
Because
The midpoint of
Therefore, the line that passes through
Solution 2:
The line through
Since the line passes through
The line with equation
First, we find the coordinates of the point at which the lines with equations
Equating values of
When
Thus, these two lines intersect at
Since all three lines pass through the same point, the line with equation
Therefore,
Suppose that
From the given information,
The integer
Why is this the only possible value of
We note that we cannot have
Thus,
Since
Since
Thus,
Since
If
Therefore, we must have
Since Eleanor has 100 marbles which are black and gold in the ratio
When more gold marbles are added, the ratio of black to gold is
Eleanor now has
First, we see that
This means that
Since
The expression
Since
First, we note that a triangle with one right angle and one angle with measure
This is because the measure of the third angle equals
In particular,
Since
Join
Consider quadrilateral
This means that
Since
This means that
Finally,
We apply the process two more times:
Before Step 1 | 24 | 3 |
---|---|---|
After Step 1 | 27 | 3 |
After Step 2 | 81 | 3 |
After Step 3 | 81 | 4 |
Before Step 1 | 81 | 4 |
---|---|---|
After Step 1 | 85 | 4 |
After Step 2 | 340 | 4 |
After Step 3 | 340 | 5 |
Therefore, the final value of
The parabola with equation
Here, the disciminant equals
The inequality
Since
(If
Since
Since
Since the difference between
Therefore, we have
We multiply both sides of the left inequality by
From this, we see that
Since
We multiply both sides of the right inequality by
From this, we see that
Since
Since
The first 6 terms of a geometric sequence with first term 10 and common ratio
Here, the ratio of its 6th term to its 4th term is
The first 6 terms of an arithmetic sequence with first term 10 and common difference
Here, the ratio of the 6th term to the 4th term is
Since these ratios are equal, then
Let
To calculate
By definition,
The prime numbers between 20 and 30, inclusive, are 23 and 29, so
Thus,
By definition,
The prime numbers between 2 and 12, inclusive, are 2, 3, 5, 7, 11, of which there are 5.
Therefore,
Since
Suppose that
From the second of these equations,
Therefore, if
Suppose that
If
If
Therefore, if
In summary, the solutions to the system of equations are
Draw a perpendicular from
Since
Therefore,
Join
Consider right-angled
Also,
By the Pythagorean Theorem,
Since
Since
Since
Now consider
Using the cosine law, we obtain the following equivalent equations:
Solution 1:
Since the function
This means that
Therefore,
This means that
Therefore,
Therefore,
Solution 2:
Since the function
To find a formula for
We are given that
We can apply the function
Thus, we can replace
Thus,
Solution 1:
Using logarithm laws, the given equations are equivalent to
Since
Since
Since
Therefore,
Solution 2:
First, we note that
From the given equations, we obtain
Since
Since
Since
Since
Therefore,
Solution 1:
Let
Because Alain, Bianca and Chen are equally matched and because their roles in the tournament are identical, then the probability that each of them wins will be the same.
Thus, the probability that Alain wins the tournament is
Let
Since exactly one of Alain, Bianca, Chen, and Dave wins the tournament, then
In order for Dave to win the tournament, he needs to win two matches.
No matter who Dave plays, his probability of winning each match is
Thus, the probability that he wins his two consecutive matches is
Thus, the probability that Bianca wins the tournament is
(We could rewrite this as
Solution 2:
Let
There are three possible pairings for the first two matches:
Bianca versus Alain, and Chen versus Dave
Bianca versus Chen, and Alain versus Dave
Bianca versus Dave, and Alain versus Chen
Each of these three pairings occurs with probability
In (i), Bianca wins either if Bianca beats Alain, Chen beats Dave, and Bianca beats Chen, or if Bianca beats Alain, Dave beats Chen, and Bianca beats Dave.
Since Bianca beats Alain with probability
Since Bianca beats Alain with probability
Therefore, the probability of Bianca winning, given that possibility (i) occurs, is
In (ii), Bianca wins either if Bianca beats Chen, Alain beats Dave, and Bianca beats Alain, or if Bianca beats Alain, Dave beats Alain, and Bianca beats Dave.
The combined probability of these is
In (iii), Bianca wins either if Bianca beats Dave, Alain beats Chen, and Bianca beats Alain, or if Bianca beats Dave, Chen beats Alain, and Bianca beats Chen.
The combined probability of these is
Therefore,
Throughout this solution, we will mostly not include units, but will assume that all lengths are in kilometres, all times are in seconds, and all speeds are in kilometres per second.
We place the points in the coordinate plane with
We put
Suppose that
Since the sound arrives at
Thus, the distance from
Since the sound arrives at
Since the distance from
Since the distance from
Since the distance from
When these equations are expanded and simplified, we obtain
After each round, each L shape is divided into 4 smaller L shapes.
This means that the number of L shapes increases by a factor of 4 after each round.
After 1 round, there are 4 L shapes.
After 2 rounds, there are
After 3 rounds, there are
There are four orientations of L shapes of a given size: ,
,
,
.
When an L of each orientation is subdivided, the following figures are obtained:
From these figures, we can see that after each subsequent round,
Each produces 2
, 0
, 1
, and 1
of the smallest size.
Each produces 0
, 2
, 1
, and 1
.
Each produces 1
, 1
, 2
, and 0
.
Each produces 1
, 1
, 0
, and 2
.
After 1 round, there are 2 , 0
, 1
, and 1
.
After 2 rounds, the number of L’s of each orientation are as follows:
:
:
:
:
After 3 rounds, the number of L’s of each orientation are as follows:
:
:
:
:
Where do these numbers come from?
For example, to determine the number of after 2 rounds, we look at the number of L’s of each orientation after round 1 (2, 0, 1, 1) and ask how many
each of these produces at the next level. Since the four types each produce 2, 0, 1, and 1
, then the total number of
after 2 rounds equals
As a second example, to determine the number of after 3 rounds, we note that after 2 rounds the number of L’s of the four different orientations are 6, 2, 4, 4 and that each L of each of the four types produces 0, 2, 1, 1
. This means that the total number of
after 3 rounds is
Putting all of this together, the number of L’s of the smallest size in the same orientation as the original L is 20.
In (b), we determined the number of L’s of the smallest size in each orientation after 1, 2 and 3 rounds.
We continue to determine the number of L’s of the smallest size after 4 rounds.
After 4 rounds, the number of L’s of each orientation are as follows:
:
:
:
:
This gives us the following tables of the numbers of L’s of the smallest size in each orientation after 1, 2, 3, and 4 rounds:
After Round | ||||
---|---|---|---|---|
1 | 2 | 0 | 1 | 1 |
2 | 6 | 2 | 4 | 4 |
3 | 20 | 12 | 16 | 16 |
4 | 72 | 56 | 64 | 64 |
We re-write these numbers in the third row as
Based on this, we might guess that the numbers of L’s of the smallest size in each orientation after
If this guess is correct, then, after 2020 rounds, the number of L’s of the smallest size in the same orientation as the original L is
We prove that these guesses are right by using an inductive process.
First, we note that the table above shows that our guess is correct when
Next, if we can show that our guess being correct after a given number of rounds implies that it is correct after the next round, then it will be correct after every round. This is because being correct after 4 rounds will mean that it is correct after 5 rounds, being correct after 5 rounds will mean that it is correct after 6 rounds, and so on to be correct after any number of rounds.
Suppose, then, that after
After
:
:
:
:
Since
Therefore, after 2020 rounds, the number of L’s of the smallest size in the same orientation as the original L is
Here, the pairwise sums of the numbers
The six pairwise sums of the numbers in the list can be expressed as
Similarly, the largest sum must be the sum of the two largest numbers, and so
Since
Since
, then .
Since, then .
Sinceand , then .
Since, then .
Thus,
Using a similar argument,
So far, we have
This means that
It turns out that either order is possible.
Case 1:
Here,
Adding these three equations gives
Since
Since
Since
Since
Thus,
Case 2:
Here,
Using the same process,
From this, we obtain
Therefore, Kerry’s two possible lists are
Suppose that the values of
In terms of the numbers
Using a similar analysis to that in (a), the smallest sum is
Also, the second smallest sum will be
We let
Even though we do not know the order in which these pairwise sums are assigned to
In other words,
Thus,
This means that
Since the values of
Using the value of
Finally, using
Therefore, given the ten sums
Suppose that the lists
Let
Consider also the list with 8 entries:
Thus, the 28 pairwise sums in each case are the same. In each case, there are 6 sums in (i), 6 sums in (ii), and 16 sums in (iii).
If we choose the initial lists to have the same pairwise sums and choose the value of
For example, if we choose
Therefore, setting