Wednesday, April 13, 2016
(in North America and South America)
Thursday, April 14, 2016
(outside of North American and South America)
©2016 University of Waterloo
The total score for School A is found by adding the scores of the four students competing from the school.
Therefore, the total score for School A is
Since the total score for School A is 36, then the total score for School B is also 36.
The scores of the four students competing from School B are
The score for Student 4 at School C,
Thus,
The scores of the four students competing from School C are
Since the total score for School C is also 36, and
Therefore, the score for Student 3 at School C is 4, and the score for Student 4 at School C is
Every 2 seconds, Esther takes 5 steps and each of her steps is 0.4 m long.
Therefore, in 2 seconds Esther travels a distance of
Solution 1
Every 2 seconds, Paul takes 5 steps and each of his steps is 1.2 m long.
Therefore, every 2 seconds Paul travels a distance of
Since Paul travels 6 m every 2 seconds, his speed is
Solution 2
Paul takes 5 steps every 2 seconds and so Paul takes 2.5 steps every second.
The length of each of Paul’s steps is 1.2 m, and so every second Paul travels a distance of
Thus, Paul travels at a speed of 3 m/s.
Solution 1
Paul travels at a speed of 3 m/s, and so in 120 seconds (2 minutes), Paul travels a distance of
Esther travels 2 m every 2 seconds and so she travels at a speed of 1 m/s.
In 120 seconds, Esther travels a distance of
If Paul and Esther start a race at the same time, then after 2 minutes, Paul will be
Solution 2
Each of Paul’s steps is
Every 2 seconds, Paul and Esther each take 5 steps, and so in
If Paul and Esther start a race at the same time, then after 2 minutes Paul will be
Solution 3
Paul travels at a speed of 3 m/s, and Esther travels 2 m every 2 seconds, so she travels at a speed of 1 m/s.
This means that in 1 second, Paul travels 3 m and Esther travels 1 m.
Every second, Paul travels
If Paul and Esther start a race at the same time, then after 120 seconds (2 minutes), Paul will be
Solution 1
Esther travels 2 m every 2 seconds or 1 m/s, and so in 180 seconds (3 minutes), Esther travels
Paul travels at a speed of 3 m/s, which is 2 m/s faster than the speed at which Esther travels.
Therefore, every second, Paul travels 2 m farther than Esther travels.
Since Esther begins the race 180 m ahead of Paul, it will take Paul
Solution 2
Esther travels 2 m every 2 seconds or 1 m/s, and so in 180 seconds (3 minutes), Esther travels
Each of Paul’s steps is
Since Paul and Esther each step at the same rate (5 steps every 2 seconds), then it will take Paul
Paul takes 5 steps every 2 seconds, and so it will take Paul
Solution 1
In
Since
In
Thus,
Solution 2
In
Since
In
Similarly,
Solution 3
In
A median of
In
, is a median and so is the midpoint of .
Therefore,and have equal bases ( ).
Further, the height ofis equal to the height of (both are ).
Thus,and have equal bases and equal heights.
Since the area of each triangle equals one-half times the base times the height, thenand have equal areas and so median divides into equal areas.
Since
Solution 1
In
Thus, by the Pythagorean Theorem,
Since
In
Thus, by the Pythagorean Theorem,
In
In
Thus,
Solution 2
Since
In
Thus,
The median of a triangle divides the area of the triangle in half.
(Solution 3 to (a) shows an example of why a median divides a triangle’s area in half.)
Since
We use the notation
In
(Solution 3 to (a) shows an example of why a median divides a triangle’s area in half.)
In
Therefore,
Since
Now
Each of these four triangles is right-angled.
Since
Since
Since
Since
Therefore,
Because the entries from one column do not affect the possible entries in another column, then the smallest possible sum of the numbers in a row equals the sum of the smallest possible number in each column.
Column | Possible entries | Smallest possible entry |
---|---|---|
B | 1 | |
I | 16 | |
N | 0 | |
G | 46 | |
O | 61 |
Thus, the smallest possible sum of the numbers in a row on a BINGO card is equal to
Solution 1
From part (a), the smallest possible sum of the numbers in a row is 124.
This minimum row sum occurs in the 3
(Note that if we do not use the 3
The minimum sum of the numbers in a diagonal is also 124 since the smallest possible number in each column (including the middle entry 0) may be used in a diagonal sum.
The minimum row sum and the minimum diagonal sum each use the numbers
However, the number 1 cannot occur in both the 3
The smallest two numbers that can appear in the 3
Similarly, the smallest two numbers that can appear in the 3
In column N, the 3
In column G, the number 46 cannot appear in both the 3
Similarly, the smallest two numbers that can appear in the 3
Therefore, in any BINGO card, the combined list of numbers in a diagonal sum and a row sum must include 10 numbers that are at least as large as 1, 2, 16, 17, 0, 0, 46, 47, 61, 62.
The sum of these numbers is
Thus, if Carrie’s BINGO card has a row and a diagonal each with the same sum, then this sum must be at least one-half of this total; that is, the minimum such sum is
One BINGO card showing this minimum equal row and diagonal sum of 126 is possible, is shown below.
B | I | N | G | O |
---|---|---|---|---|
1 | ||||
17 | ||||
2 | 16 | 0 | 47 | 61 |
46 | ||||
62 |
We confirm that the row and diagonal sums are
Each of the empty spaces in the card may be filled with any of the appropriate numbers not yet used.
Solution 2
From part (a), the smallest possible sum of the numbers in a row is 124.
This minimum row sum occurs in the 3
(Note that if we do not use the 3
The minimum sum of the numbers in a diagonal is also 124 since the smallest possible number in each column (including the middle entry 0) could be used in a diagonal sum.
The minimum row sum and the minimum diagonal sum each must use the numbers
However, the number 1 cannot occur in both the 3
Similarly, 16 cannot occur in both the 3
A row sum of 126 and a diagonal sum of 126 is possible, as the following BINGO card shows:
B | I | N | G | O |
---|---|---|---|---|
1 | ||||
16 | ||||
2 | 17 | 0 | 46 | 61 |
47 | ||||
62 |
We confirm that the row and diagonal sums are
Is it possible to have a BINGO card with a row sum of 125 and a diagonal sum of 125?
If not, then the smallest possible number that can be both a row sum and a diagonal sum will be 126.
Consider the smallest possible diagonal sum
Since 1, 16, 0, 46, 61 are the smallest possible entries in each column, then a diagonal sum of 125 can only be created by replacing exactly one of the four integers 1, 16, 46, 61 by the integer that is one larger.
In particular, the possible diagonals with a sum of 125 are
It is not possible for a BINGO card to have one of these sums as a diagonal sum and a different one of these sums as a 3
Therefore, the smallest possible number that can be both a diagonal sum and a row sum is 126.
Solution 1
The maximum possible sum of the numbers in the 3
We need the sum of the numbers in the 3
We determine the number of ways in which this can be done by starting with the largest possible numbers and reducing these numbers to reduce the sums to 177.
Thus, we call the missing numbers in the 3
B | I | N | G | O |
---|---|---|---|---|
23 | 35 | 47 | 65 | |
5 | 31 | 52 | 63 | |
0 | ||||
11 | 20 | 40 | 69 | |
9 | 18 | 38 | 48 |
Since the numbers in the first column are between 1 and 15, inclusive, then
Since the numbers in the 3
or
Similarly, we have
Since the B column cannot contain repeated numbers, then
The number of BINGO cards with the desired property is equal to the number of ways that we can choose non-negative integers
Since
For the sum of non-negative integers to be 3, no single integer can be larger than 3. If one integer is 3, the rest are 0. If one integer is 2, then we must have one 1 and the rest equal to 0. If no integers equal 3 or 2, then we must have three 1s.
Similarly,
Note that since none of the values can be larger than 3, then the missing entries in the B, I, G, and O columns are at least 12, 27, 57, and 72, respectively, so cannot duplicate existing entries.
To count the BINGO cards, we now count the possible combinations of values for
In this discussion, we call
Case 1: W, X, Y, Z are 3, 0, 0, 0 in some order
Since
Looking at
In total, this means that there are 4 possible ways in which this can happen.
Case 2: W, X, Y, Z are 2, 1, 0, 0 in some order
Looking at
Looking at
Overall, there are
In how many ways can this happen?
Looking at
Looking at
Overall, there are
Case 3: W, X, Y, Z are 1, 1, 1, 0 in some order
In total, there are thus
Each set of values of these variables gives a BINGO card with the desired property, and so there are
Solution 2
The maximum possible sum of the numbers in the 3
We require both the sum of the numbers in the 3
Since 177 is 3 less than the maximum possible sum of 180, then any of the missing numbers in the given BINGO card can be at most 3 less than the largest number that can appear in columns B, I, G, and O (column N is fixed at 0).
That is, the smallest number that can appear in the 3
Similarly, the smallest numbers that can appear in the 3
Thus, the missing numbers in the given BINGO card must be chosen from:
(Note that these numbers do not already appear in the given BINGO card and so they each may be chosen to fill blank spaces.)
There are three different methods in which the maximum sum, 180, can be decreased by exactly 3 to give a row or diagonal sum of 177.
From the lists above, we may choose:
the smallest number from one of the four columns (this number is 3 less than the largest), and choose the largest number from each of the remaining three columns. For example we could choose the smallest number from column B, 12, and the largest numbers from the remaining columns,
the largest number from one of the four columns, and choose the second largest number from each of the remaining three columns. For example we could choose the largest number from column B, 15, and the second largest numbers from the remaining columns,
the largest numbers from two of the four columns, and choose the second largest number from one of the remaining two columns and the third largest number from the final column. For example we could choose the largest numbers from columns B and I, 15 and 30, and the second largest number from column G, 59, and the third largest number from column O, 73, since
These are the only three methods in which we can decrease the maximum row and diagonal sum of 180 by exactly 3 to give a row or diagonal sum of 177.
We restate these three methods by considering the following table:
B | I | G | O | |
---|---|---|---|---|
P | 15 | 30 | 60 | 75 |
Q | 14 | 29 | 59 | 74 |
R | 13 | 28 | 28 | 73 |
S | 12 | 27 | 57 | 72 |
The three methods for achieving a row or diagonal sum of 177 are:
choose 1 number from group
choose 1 number from group
choose 2 numbers from group
We require both the 3
Thus, we must use two of the above methods simultaneously and we must ensure that no number appears twice in any given column.
Which pairs of combinations may be chosen from the three methods listed?
If we fill the blanks in the 3
Further, if we fill the blanks in the 3
Therefore, if the 3
Similarly, we cannot fill the blanks in the 3
We also cannot fill the blanks in the 3
All other combinations of methods are possible and they are summarized in the table:
Method used to fill the 3 |
Method used to fill the diagonal |
---|---|
1: |
2: |
2: |
1: |
2: |
3: |
3: |
2: |
3: |
3: |
Finally, to count the number of ways to complete the BINGO card, we must count the number of ways to choose numbers that satisfy each of the five combinations listed above.
First Combination: SPPP in the 3rd row and PQQQ in the diagonal
Select one of the 4 columns (B, I, G, O) in which to choose the group
number, and the remaining 3 columns are each filled with their group number.
To complete the diagonal in this same BINGO card using
That is, there is only 1 choice for the placement of the group
So there are 4 ways to select the numbers for the 3
Second Combination: PQQQ in the 3rd row and SPPP in the diagonal
The counting here is identical to that of the first combination above, with the roles of the
Thus, there are 4 ways in which the BINGO card can be completed using this second combination.
Third Combination: PQQQ in the 3rd row and PPQR in the diagonal
Select one of the 4 columns (B, I, G, O) in which to choose the group
number, and the remaining 3 columns are each filled with their group number.
To complete the diagonal in this same BINGO card using
Next we must fill the remaining 3 columns with their respective
These can be ordered in 3 different ways (
So there are 4 ways to select the numbers for the 3
Fourth Combination: PPQR in the 3rd row and PQQQ in the diagonal
The counting here is identical to that of the third combination above, with the roles of the
Thus there are 12 ways in which the BINGO card can be completed using this fourth combination.
Fifth Combination: PPQR in the 3rd row and PPQR in the diagonal
Select one of the four columns in which to place the group
number. There are 4 choices for this column, and for each of these choices, there are 3 choices of column in which to place the group number. The group numbers are then placed in the empty columns.
We then need to place the numbers on the diagonal. There are 2 ways to do this:
The group
numbers on the diagonal must go in the columns corresponding to the locations of the group and numbers in the 3 row.
The groupand numbers on the diagonal can be placed in the two remaining columns in 2 ways – either or when reading from left to right.
Therefore, there are
In total, the number of ways to complete the BINGO card so that the sum of the numbers in the diagonal is 177, and the sum of the numbers in the 3