Wednesday, November 22, 2017
(in North America and South America)
Thursday, November 23, 2017
(outside of North American and South America)
©2017 University of Waterloo
Calculating,
Answer: 5
Solution 1:
Since
Since
We are told that the area of
Therefore,
Solution 2:
Since
Since
Since rectangle
Therefore,
Answer: 4.8 cm
Factoring, we obtain the following equivalent equations:
Therefore,
Answer:
Solution 1:
Since the two 1s are not next to each other, then the two 1s can be placed in the following pairs of positions, reading from the left: 1st and 3rd, 1st and 4th, 1st and 5th, 2nd and 4th, 2nd and 5th, 3rd and 5th.
There are 6 such pairs of positions.
Choose one of these pairs, say 1st and 3rd. This gives the number
There are 3 digits left to place. We place these from left to right.
There are 3 possible digits that could go in the leftmost empty position.
After this digit is placed, there are 2 possible digits that could go in the next empty position.
Finally, the remaining digit is placed in the remaining empty position.
This process works for each of the pairs of positions for the two 1s.
Therefore, there are
Solution 2:
Since there are 5 digits to arrange and 2 of them are the same, there are
To see this, replace one of the 1s with and
so that the “digits” were . There would be 5 choices for the first digit, 4 choices for the second digit, and so on, giving arrangements of the digits.
If we now replace thewith a 1, each arrangement is now counted 2 times. For example, and become the same arrangement.
Therefore, we need to divide the totalby since each arrangement is double-counted.
In some of these arrangements, the 1s will be next to each other and in some they will not be.
We will count the arrangements with the 1s next to each other and subtract this number from the total.
If the two 1s are together, we can imagine arranging the four objects 11, 2, 3, 4.
There are
Thus, there are
Answer: 36
We note that the first point
In the diagram, line segments of lengths
We guess that the points on the spiral that lie on the line
To see why this is true, we note that, after
Note that
We note further that no more points on the spiral can lie on the line
At the point after
We note that the expression is increasing as
Therefore, we want to calculate the sum of the values of
We obtain
Therefore, the sum of the positive integers from 1 to 1000 written at points on the line
Answer: 10 944
Since
We label the triangle as
We put the diagram in the
Let
Since
Note also that since
This also means that the coordinates of
Since
This also means that the coordinates of
Since
The coordinates of
Let
Let the points
Join
It turns out that
Imagine drawing a line tangent to the large circle at
.
Since the large circle and semi-circle touch at, then this line is also tangent to the semi-circle.
Since radii are perpendicular to tangents, thenand (both radii) are perpendicular to this tangent line.
Since bothand are perpendicular to the tangent line, then they must lie on top of each other, and so passes through .
Using a similar argument, we can see thatand pass through and , respectively.
Since
Using the formula for the length of a line segment, we obtain the equations
Subtracting the third equation from the second, we obtain
Substituting
Answer:
Factoring the left side of
Therefore, the roots of
Since the parabola with equation
Therefore,
Subtracting the first of these equations from the second, we obtain
Since
Therefore,
(We can check that the points
Since the point
Thus, the parabola has equation
We want to determine the value of
Substituting, we obtain the following equivalent equations:
Since
The following table shows how Joe can win the game in three turns:
Start | R | R | R | R | R | R |
---|---|---|---|---|---|---|
After 1 turn | G | G | G | G | R | R |
After 2 turns | G | R | R | R | G | R |
After 3 turns | G | G | G | G | G | G |
On Joe’s first turn, he turns over the first 4 cards.
On Joe’s second turn, he turns over cards 2 through 5.
On Joe’s third turn, he turns over the four red cards.
There are many sequences of moves in which Joe can win.
Suppose that Joe takes 9 turns. On turn 1, he turns over cards 1, 2, 3, 4, 5. On turn 2, he turns over cards 2, 3, 4, 5, 6.
He continues in this way so that, on each turn, he turns over five consecutive cards starting with the
In this way, each of the 9 cards is turned over 5 times (once as each of the 1st, 2nd, 3rd, 4th, 5th card in the sequence).
Since each card is turned over an odd number of times, its final colour is the opposite of the starting colour, and so it is green.
We demonstrate this in the following chart:
Start | R | R | R | R | R | R | R | R | R |
---|---|---|---|---|---|---|---|---|---|
After 1 turn | G | G | G | G | G | R | R | R | R |
After 2 turns | G | R | R | R | R | G | R | R | R |
After 3 turns | G | R | G | G | G | R | G | R | R |
After 4 turns | G | R | G | R | R | G | R | G | R |
After 5 turns | G | R | G | R | G | R | G | R | G |
After 6 turns | R | R | G | R | G | G | R | G | R |
After 7 turns | G | G | G | R | G | G | G | R | G |
After 8 turns | R | R | R | R | G | G | G | G | R |
After 9 turns | G | G | G | G | G | G | G | G | G |
Joe can actually finish in as few as three turns:
Start | R | R | R | R | R | R | R | R | R |
---|---|---|---|---|---|---|---|---|---|
After 1 turn | G | G | G | G | G | R | R | R | R |
After 2 turns | G | G | R | R | R | G | G | R | R |
After 3 turns | G | G | G | G | G | G | G | G | G |
Suppose that
We show that Joe can win the game when
Suppose that
Suppose that Joe takes 2017 turns.
For each
In this way, each of the 2017 cards is turned over
Since
In this way, Joe wins the game when
Suppose that
For Joe to win the game, each of the 2017 cards must be turned over an odd number of times in order to reverse its colour.
This means that the total number of card flips is odd, since this total is the sum of 2017 odd integers (the number of flips for each of the 2017 cards).
For any positive integer
Since
Therefore, after any number of turns, the total number of card flips is always even and so cannot be the odd number of flips necessary to reverse the colour of all of the cards.
Therefore, when
In summary, when
Continuing to substitute values of
We show that there is no such function
At each step, we use values of
Assume that there exists a function
If
If
If
If
Since
If
Since
Therefore, there does not exist such a function
Solution 1:
First, we show that the function
We note that this function satisfies the first two conditions.
Thus, if
Next, we show that there is no other such function.
Suppose that the function
Step 1: Show that
Using the functional equation with
Using the functional equation with
Using the functional equation with
Assume that
Using the functional equation with
Step 2: Show that if
In other words, we show that the function
Suppose that
Then for each positive integer
Since
Step 3: Show that
From Step 1,
Suppose that
When
The values of
When
The values of
When
The values of
Since
But the three sets of outputs
This is a contradiction, so we cannot have
Step 4: Show that
We know that
If
Substituting
In this case,
Therefore, we cannot have
Step 5:
From
Step 6:
From
If we can show that
Since
In particular,
From
When
This is because
Therefore,
Since
This means that we must have
If
If
Therefore,
We have shown that the function
Solution 2:
Let
Let
When
Thus, the point
When
Thus, the point
Suppose that
When
Thus, if the point
This shows that the infinite set of points of the form
Note that to get from one point in this set to the next, we move
In other words, this infinite set of points on the graph forms a line with slope
Call this line
Suppose that
A similar argument shows that the set of points
Now consider the points
There are infinitely many such points (for example,
These points lie on both
But this is true for every possible pair of distinct positive integers
Therefore, the quantity
In other words,
If
Setting
Since
Since
(Solution 1 shows us that the function