Wednesday, April 15, 2020
(in North America and South America)
Thursday, April 16, 2020
(outside of North American and South America)
©2020 University of Waterloo
Locating point
From part (a), Daniel pays $1.60 per poster and Cao pays $0.50 per poster.
Calculating, Annie pays $10.00 for 5 posters or
Thus, Bogdan and Emily are paying the same price per poster.
(Alternatively, we may have noticed that the line from the origin
The slope of this line represents the price in dollars per poster for Bogdan and for Emily.)
From part (a), Daniel paid the company that printed his first batch $1.60 per poster.
To print his second batch at the local library, Daniel would pay
To spend less money, Daniel should print his second batch at the library.
Alternatively, Daniel paid $16.00 for his first batch of 10 posters and thus would pay
Since the cost to print the 40 posters at the library is $60.00, he should print them at the library to save money.
In part (b), we calculated that Emily’s printing company charges her $1.00 per poster.
Since this is a fixed price per poster, then Emily paid $1.00 per poster for each of the 25 posters that she had printed, for a total of $25.00 spent.
Annie and Emily each printed 25 posters and spent the same amount of money, $25.00.
Annie paid $10.00 to print her first 5 posters, and so she spent
Thus, Annie was charged
In
We begin by showing that
Since
In addition, hypotenuse
Thus,
Since
Since
Using the Pythagorean Theorem in
Thus the height of
We begin by determining the area of
Construct the height of
Since
By symmetry,
The area of
The area of
The fraction of the area of
If the
To determine the
To see this, consider that if two consecutive terms in a Dlin sequence are
To determine the operations needed to find
(We may check that the term following 70 is indeed
If the
If the
If the
If the
At this point, we see that 174, 350, 702, and 1406 are possible
We may continue this process of working backward (dividing by 2 and subtracting 1) to determine all possible
Thus, the possible
Each of the integers from 10 to 19 inclusive is a possible first term, and so we must determine the ones digit of each term which follows each of these ten possible first terms.
If the
If the
Given each of the possible first terms, we list the ones digits of the
1 |
10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
---|---|---|---|---|---|---|---|---|---|---|
Units digit of the 2 |
2 | 4 | 6 | 8 | 0 | 2 | 4 | 6 | 8 | 0 |
Units digit of the 3 |
6 | 0 | 4 | 8 | 2 | 6 | 0 | 4 | 8 | 2 |
From the table above, we see that the only ones digit which repeats itself is 8.
Thus, if the
Similarly, if the
It then follows that all further terms after the first will have ones digit 8.
The
If the
For example, if
What is the largest possible value of
Setting
Since the
Similarly, solving
When
Thus, if a Dlin sequence has
Further, 503 is the largest possible
Each
Thus, to count the number of positive integers between 1 and 2020, inclusive, that can be the
The smallest possible
Further, every value of
Thus, there are 503 positive integers between 1 and 2020, inclusive, that can be the
There are 10 different ways to colour a
One way to see this is to count the number of different ways to colour 2 cells blue, since each of the 3 remaining cells must be coloured red.
(Alternatively, we could count the number of different ways to colour 3 cells red.)
There are 5 cells that may be chosen first to be coloured blue.
After this chosen cell is coloured, there are 4 remaining cells that can be chosen to be coloured blue. Thus, there are
However, since the two blue cells are indistinguishable from one another, we have counted each of the different ways of colouring 2 cells blue twice.
For example, choosing to first colour the cell in the second row blue and then colouring the cell in the fifth row blue gives the same colouring as first choosing to colour the cell in the fifth row blue and then colouring the cell in the second row blue.
Thus, there are
We begin by observing that there are exactly 2 different ways to colour each of the 13 cells (red or blue), and so there are a total of
For each
Since there are 13 cells in total, then
If
If
Thus, if the number of cells coloured red is greater than the number of cells coloured blue, then Carrie writes down the number of red cells (since she writes the maximum of
Alternatively, if the number of cells coloured blue is greater than the number of cells coloured red, then Carrie writes down the number of blue cells, and this number is at least 7.
In either case, each number that Carrie writes down in her list is between 7 and 13, inclusive.
At least one of the possible
Since Carrie’s list includes a 7 and every number in her list is 7 or greater, then the smallest number in her list is 7.
In a
Each of these 3 cells can be coloured in one of two ways, either red or blue.
Since there are two ways that each of the 3 cells may be coloured, each column in a
Thus, it is possible to colour a
Since there are only 8 different ways to colour the 3 cells in any column, then every colouring of a
Thus, the smallest possible value of
The given statement is true.
In a
In each column, at least 3 of the 5 cells must be the same colour.
In each column, either there are more cells coloured red than blue or there are more cells coloured blue than red (they can’t be equal in number since 5 is odd).
In each column, if there are more cells coloured red than blue, we call that column a red column.
Alternatively, if there are more cells coloured blue than red, we call that column a blue column.
Let the total number of red columns be
Similar to the argument in part (b), if
Assume that
In this case, each of these 21 (or more) columns has more cells coloured red than those that are coloured blue and so each of these columns has at least 3 cells coloured red.
We will show that the location of 3 cells coloured red is the same in at least 3 of 21 columns.
Of all red columns, consider the first 21 (there are at least 21).
Moving from top to bottom within each of these red columns, consider the first 3 red cells (there are at least 3 cells coloured red).
What is the maximum number of ways to colour each column so that exactly 3 cells are coloured red?
Since a
In the 21 red columns, assume that at most 2 columns have the same 3 cells coloured red.
Since there are only 10 different ways to colour each column so that exactly 3 cells are coloured red, then there are at most
However, there are 21 red columns and so we have a contradiction.
Thus, our assumption that in the 21 red columns, there are at most 2 columns in which the same 3 cells are coloured red was incorrect, and so there must be at least 3 columns which have the same 3 cells that are coloured red.
The 9 cells located at the intersection of these 3 columns and the 3 rows containing the red cells within these columns are all coloured red.