Wednesday, November 18, 2018
(in North America and South America)
Thursday, November 19, 2018
(outside of North American and South America)
©2018 University of Waterloo
Since the angles in any triangle add to
Answer:
Since 10% of the 500 animals are chickens, then
The remaining
Since there are twice as many goats as cows, then the total number of goats and cows is three times the number of cows.
Therefore, there are
(Since there are 150 cows, then there are
Answer: 150 cows
Since
Therefore,
Since
Furthermore,
Since
Answer:
Let
Since there are 8 numbers across the bottom row of this square, then the number in the bottom left corner is
In the larger grid, each number is 24 greater than the number directly above it.
The number in the bottom right corner of the
Since there are 24 numbers in each row, each number in the grid above the bottom row is 24 less than the number directly below it.
This means the number in the top right corner is
Since the number in the bottom right corner is
Moving to the left across the first row of the
Since the sum of the numbers in the four corners of the square is to be
We should also verify that, starting with 499 in the bottom right corner, we can construct an
Since the bottom right number in the grid is 576, then the numbers up the right side of the grid are 576, 552, 528, 504, which means that, moving from right to left, 499 is sixth number of the 4th row from the bottom.
Since the grid is
Answer: 499
Consider the point
Here,
Because
Therefore, the point
If a point
In other words, any point
To get from
Therefore, starting at
Similarly, the point
In other words, the triangle with vertices at
(To do this more formally, we could determine that the equation of the line through
Next, consider
Again, the area of
Moving right 3 and up 4 from
Since we are told that there are five such points, then we have found all of the points and they are
(We could note that any point
Answer:
We know that
We focus on the possible values of
Consider
Since there are 20 chairs, then moving 20 chairs around the circle moves each person back to their original seat.
Therefore, any configuration of people sitting in chairs is preserved by this movement.
In other words,
Thus, there are 20 pairs
Consider
Here, any people in chairs 1 to 10 move to chairs 11 to 20 and any people in chairs 11 to 20 move to chairs 1 to 10.
This means that the two halves (1 to 10 and 11 to 20) must contain the same number of people in the same configuration. (That is, if there are people in chairs 1, 3, 4, 8, then there are people in chairs 11, 13, 14, 18, and vice versa.) This means that the total number of people in chairs is even.
There are 10 possibilities for the number of chairs occupied among the first 10 chairs (1 to 10), and so 10 possibilities for the total number of chairs occupied (the even numbers from 2 to 20, inclusive).
Thus, there are 10 pairs
Consider
Here, any people in chairs 1 to 5 move to chairs 6 to 10, any people in chairs 6 to 10 move to chairs 11 to 15, any people in chairs 11 to 15 move to chairs 16 to 20, and any people in chairs 16 to 20 move to chairs 1 to 5.
This means that the four sections containing 5 chairs must contain the same number of people in the same configuration.
This means that the total number of people in chairs is a multiple of 4 and can be 4, 8, 12, 16, or 20, which gives 5 possible values of
Thus, there are 5 pairs
If we consider
Thus, there are 4 pairs
Consider
Here, each person in a chair moves 1 chair along the circle.
If there is a person in chair 1, then they move to chair 2, which means that there must have been a person in chair 2. This person has moved to chair 3, which means that there must have been a person in chair 3, and so on.
Continuing, we see that all 20 chairs must be full, and so
Thus, there is 1 pair
Consider
In each of these cases,
If
A similar cycle can be constructed for
Consider
Looking at the cycles as in the previous case,
This means that each set of 10 chairs are either all occupied or are all not occupied.
This means that there are two possible values of
Thus, there are 2 pairs
Consider
Here, there are 4 cycles:
Note that starting at any chair other than 1, 2, 3 or 4 gives one of these four cycles. That is, all 20 chairs appear here.
Thus, there are 4 pairs
Finally, we consider
Moving 11 seats clockwise gives the same result as moving 9 seats counterclockwise. Since the arguments above do not depend on the actual direction, then the number of pairs when
Similarly, the numbers of pairs for
This gives the following chart:
Values of |
Number of values of |
---|---|
20 | 20 |
10 | 10 |
5, 15 | 5 |
4, 8, 12, 16 | 4 |
2, 6, 14, 18 | 2 |
1, 3, 7, 9, 11, 13, 17, 19 | 1 |
Therefore, the number of pairs
Answer: 72
We arrange the list in increasing order and obtain
Therefore, the range of this list is
Removing
Since
If
If
Therefore, the two possible values of
Since
In other words, when the given list is arranged so that each term is greater than or equal to the one before it, we obtain
Since
In other words, when the given list is arranged in increasing order, we obtain
Since
(To see why there are no more solutions, we note that since
Thus,
Since there are 11 balls, then there are
Since there are 7 black balls, then there are
Therefore, the probability that both of the balls that Julio removes are black is
Since there are
Since there are 6 black balls, then there are
Since the probability that both of the balls that Julio removes are black is
By trial and error, we can determine that
Alternatively, we can expand and factor:
Since
Since there are
Since there are
Since we are given that the probability that both of the balls that Julio removes are black is
Since
Since there are
For Julio to draw two black balls and one gold ball, he could draw “black, black, gold” or “black, gold, black” or “gold, black, black”.
The number of ways in which the first possibility can happen is
The number of ways in which the second possibility can happen is
The number of ways in which the third possibility can happen is
In other words, there are
Since the probability that two of the three balls are black and one is gold is at least
Since
As
This means that we can find an integer
We note that
Since
When
When
When
Therefore,
In Figure 1,
Since
Thus,
Therefore,
Since
Extend
We have now created a new version of the configuration in Figure 1.
Since
Using the information determined so far,
Therefore,
Since
Note that each of
Since
Since
If
As in (b), we extend
We have again created a new version of the configuration in Figure 1.
Since
Using the information determined so far,
Since
Now
Since
We are given that
Substituting, we obtain successively the equivalent equations
Since
So
Factoring 2541, we obtain
We are asked for the possible lengths of
Therefore, the possible lengths for
It is indeed possible to construct a trapezoid in each of these cases. The corresponding lengths of