Wednesday, November 20, 2019
(in North America and South America)
Thursday, November 21, 2019
(outside of North American and South America)
©2019 University of Waterloo
Since
Since
Therefore,
Thus,
Answer:
Solution 1
Binh’s 20 quarters are worth
Abdul’s 20 dimes are worth
Since Binh’s and Abdul’s coins have the same total value, then the value of Abdul’s quarters is
Since each quarter is worth 25 cents, then Abdul has
Solution 2
Binh’s 20 quarters are worth
Abdul’s 20 dimes are worth
Suppose that Abdul has
Since Binh’s and Abdul’s coins have the same total value, then
Answer: 12 quarters
We note that
Since
Thus,
Answer: 41
Ali earns a total of 12 points for her 12 correct answers. To determine her possible total scores, we need to determine her possible numbers of bonus points.
Since Ali answers 3 questions incorrectly, these could all be in 1 category, split over 2 categories (2 from 1 category and 1 from another), or split over 3 categories (1 from each).
In the first case, she answers all of the questions in 4 of the 5 categories correctly, and so earns 4 bonus points. In this case, her total score would be
In the second case, she answers all of the questions in 3 of the 5 categories correctly, and so earns 3 bonus points. In this case, her total score would be
In the third case, she answers all of the questions in 2 of the 5 categories correctly, and so earns 2 bonus points. In this case, her total score would be
These are all of the possibilities.
Therefore, Ali’s possible total scores are 14, 15 and 16.
Answer:
Since
We count the number of possible pairs
Suppose that
Since
Since there is 1 possible value for
Suppose that
Since
Since there are 2 possible values for
Suppose that
Here,
Overall, there are
As
We check the final case
Suppose that
Here,
Overall, there are
In total, this means that there are
Grouping the last 10 numbers in pairs from the outside towards the middle we obtain
Thus, there are
(This problem can also be solved using a neat result called Pick’s Theorem. We encourage you to look this up and think about how you might apply it here.)
Answer:
Suppose that the original circle has radius
Thus, the circumference of this circle is
When this circle is cut into two pieces and each piece is curled to make a cone, the ratio of the lateral surface areas of these cones is
This means that the ratio of the areas of the two pieces into which the circle is cut is
In other words, the sector cut out is
Since the central angles of the two pieces are
Since the circumference of the original circle is
These pieces become the circumferences of the circular bases of the two cones.
Since the ratio of the circumference to the radius of a circle is
Since the radius of the original circle becomes the slant height of each cone, then the slant height in each cone is
In a cone, the radius and the height are perpendicular forming a right-angled triangle with the slant height as its hypotenuse.
Therefore, the height of a cone with slant height
The volume of a cone with radius
Dividing the first volume by the second, we obtain
Answer:
In the diagram,
By the Pythagorean Theorem,
Since
In the diagram,
By the Pythagorean Theorem,
Since
(Note that
Using the given fact, we can compute the ratio by computing the ratio of the differences of
We could also have this with
Therefore, the ratio of lengths
Solution 1
The difference between the
This difference is split in the ratio
The difference between the
This difference is split in the ratio
Since
Verifying, using the points
Solution 2
Suppose that
Since
From
From
Thus,
For
Since
Therefore, the point
By definition,
We first note that a two-digit number “
This means that
Since
Using our work so far,
From our earlier work,
This means that we want to find all non-zero digits
Since
Since
Case 1:
Here,
Note that
Since
Since
This means that
Since
This means that
Since
If
Also, if
There are thus 5 grids in this case:
Here,
As in Case 1,
Since
This means that
Since
This means that
Therefore,
No such integer exists, and so there are no solutions in this case.
Case 3:
Here,
As in Case 1,
Since
This means that
Since
This means that
Therefore,
No such integer exists, and so there are no solutions in this case.
Case 4:
Here,
Since
To have
There is thus 1 grid in this case:
In summary, the grids that satisfy
After 4 people, there is 1 person at the Left table, 1 person at the Centre table, and 2 people at the Right table.
Continuing the table, we obtain
Left | Middle | Right | |||
---|---|---|---|---|---|
5 | 3 | 6 | P1 | ||
5 | P2 | 3 | 3 | ||
3 | P3 | 3 | |||
3 | P4 | ||||
P5 | 2 | ||||
2 | P6 | ||||
P7 |
Person 5 sits at the Left table because
Person 6 sits at the Right table because
Person 7 sits at the Left table because
Therefore, Person 5 sits at the Left table, Person 6 at the Right table, and Person 7 at the Left table.
Suppose that there are integers
We construct a similar chart from the given information, using the information about where the previous people have sat to calculate the shares in each row:
Left | Middle | Right | |||
---|---|---|---|---|---|
P1 | |||||
P2 | |||||
P3 | |||||
P4 | |||||
P5 | |||||
P6 |
Since Person 1 sits at the Left table, then their potential share at the Left table is at least as large as it is at the Middle and Right tables.
Therefore,
Since Person 2 sits at the Middle table, then their potential share at the Middle table is larger than that at the Left table (if they were equal, they would choose Left) and at least as large as it is at the Right table.
Therefore,
Since Person 6 sits at the Left table, then their potential share at the Left table is at least as large as it is at the Middle and Right tables.
Therefore,
The inequality
But this gives
These two inequalities cannot both be true.
This means that we have a contradiction and so there are no values of
Since
Since
Since
We note also that each additional person who sits at a table causes the share per person at that table to decrease.
Since each person sits at the table that maximizes their current share of chocolate, then there cannot be 10 people at the Left table before there are 19 people at the Middle table and 25 people at the Right table, because the 10th person at the Left table would get a larger share by sitting at the Middle or Right table.
Similarly, there cannot be 20 people at the Middle table before there are 9 people at the Left table and 25 people at the Right table, and there cannot be 26 at the Right table before there are 9 people at the Left table and 19 people at the Middle table.
In other words, we must have 9 people at the Left table, 19 people at the Middle table, and 25 people at the Right table before there are more than 9, 19 and 25 people at the Left, Middle and Right tables, respectively.
At this point, there are
The largest multiple of 53 less than 2019 is
We note also that
The 342nd person to sit at the Left table would be getting a share of
The 722nd person to sit at the Middle table would be getting a share of
The 950th person to sit at the Right table would be getting a share of
Using a similar argument to that above, we cannot have more than 342 people at the Left table before there are 722 people and 950 people at the Middle and Right tables. Similarly, we cannot have more than 722 people at the Middle table before there are 342 people and 950 people at the Left and Right tables, or more than 950 people at the Right table before there are 342 people and 722 people at the Left and Middle tables.
In other words, once 2014 people have been seated, there are 342 people at the Left table, 722 people at the Middle table, and 950 people at the Right table.
To determine at which table Person 2019 sits, we first determine where Person 2015, Person 2016, Person 2017, and Person 2018 sit:
Left | Middle | Right | |||
---|---|---|---|---|---|
P2015 | |||||
P2016 | |||||
P2017 | |||||
P2018 | |||||
P2019 |
Therefore, Person 2019 sits at the left table.