Wednesday, November 18, 2020
(in North America and South America)
Thursday, November 19, 2020
(outside of North American and South America)
©2020 University of Waterloo
We write each of the three given fractions as a decimal:
Re-writing this list in increasing order, we obtain 0.04, 0.25, 0.4, 0.404, 0.41.
Therefore, the middle number is 0.4 or
Answer:
Each 8 by 10 rectangle has area
The 4 by 4 square has area
Each of the two shaded pieces is the part of an 8 by 10 rectangle outside a 4 by 4 square, and so has area
Therefore, the total area of the shaded region is
(The area of the shaded region could also be determined by dividing it into rectangular pieces.)
Answer:
After 10 days, Juan has removed
On day 11, Juan removes 11 candies so has now removed
This means that, after day 10, Juan has removed fewer than 64 candies, and, after day 11, Juan has removed more than 64 candies.
Therefore, the smallest possible value of
Answer: 11
Solution 1
The inner polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the shorter parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed between two identical trapezoids).
The sum of the measures of the angles in this polygon with 9 sides is
Thus, the measure of each of the interior angles is
Let
Since the nine trapezoids are identical, then the trapezoid adjacent to the left also has an angle of
Since the sum of the measures of angles around a point is
Solution 2
The outer polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the longer parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed by two of the base angles of two identical trapezoids.)
The sum of the measures of the angles in this polygon with 9 sides is
Thus, the measure of each of the interior angles is
Each of these interior angles is made up from two equal base angles, which are thus equal to one-half of
Since the trapezoids have parallel sides, then
Answer:
Throughout this solution, we use the fact that if
Since
Since
Suppose that
Since
This means that
Since
Since
If
If
If
If
Therefore, when
Suppose that
Since
This means that
Since
If
If
If
Therefore, when
In summary, the solutions are
Answer:
When each of 90 players plays against all of the other 89 players exactly once, the total number of games played is
This is because each of the 90 players plays 89 games, giving
Exactly 1 point is awarded in total for each game. This is because the two players either win and lose (
Therefore, over 4005 games, a total of 4005 points are awarded to the players in the league.
Suppose that, after all games have been played,
Then
Since
Suppose that 74 players have at least 54 points.
This accounts for at least
This leaves at most
These 16 players play
This means that these 16 players have at least 120 points between them, and possibly more if they obtain points when playing the other 54 players.
Therefore, these 16 players have at most 9 points and at least 120 points in total, which cannot happen.
Thus, it cannot be the case that 74 players have at least 54 points.
Suppose that 73 players have at least 54 points.
This accounts for at least
This leaves at most
These 17 players play
This means that these 17 players have at least 136 points between them.
Therefore, these 17 players have at most 63 points and at least 136 points in total, which cannot happen.
Thus, it cannot be the case that 73 players have at least 54 points.
Suppose that 72 players have at least 54 points.
This accounts for at least
This leaves at most
These 18 players play
This means that these 18 players have at least 153 points between them.
Therefore, these 18 players have at most 117 points and at least 153 points in total, which cannot happen.
Thus, it cannot be the case that 72 players have at least 54 points.
Suppose that 71 players have at least 54 points.
This accounts for at least
This leaves at most
These 19 players play
This means that these 19 players have at least 171 points between them.
These point totals match. Therefore, it appears that 71 players could have at least 54 points.
It is in fact possible for this to happen.
Suppose that every game between two of the 71 players ends in a tie, every game between one of the 71 players and one of the 19 players ends in a win for the first player, and every game between two of the 19 players ends in a tie.
In this case, each of the 71 players has 70 ties and 19 wins, for a point total of
Answer: 71
In the arithemetic sequence
This constant is
Therefore, the next two terms are
The arithmetic sequence
This means that the constant difference (that is, the constant that is added to one term to obtain the next) is added 4 times to the first term (which equals 2) to obtain the last term (which equals 14).
Thus, the constant difference must be
This means that the sequence starts at 2 and counts by 3s: 2, 5, 8, 11, 14.
Therefore,
There are six possible ways in which the three terms can be arranged:
Therefore, the possible values of
Suppose that the sequence
This means that the difference between any two consecutive terms is
Since the sequence has 6 terms, then the difference is added 5 times to
Also, the difference between a later term in the sequence and an earlier term in the sequence must equal one of
Since 4 and 20 are two terms in the sequence, with 20 appearing later than 4, and
The corresponding values of
Since
(The sequence
The sequence
The volume of Tank B is
This means that Tank B is
Since Tank B fills at
Since the volume of Tank B is
(We can also note that since it is
Since Tank A drains at
Since the volume of Tank A is
Tank A is sitting on one of its
Since the water itself forms a rectangular prism, the depth of water in Tank A at this instant is
Since Tank A starts full and Tank B starts empty and the rate at which water leaves Tank A equals the rate at which water enters Tank B, then the combined volume of water in the two tanks is constant and equal to the initial volume of Tank A, or
Let the depth of water in Tank A and Tank B when their depths are equal be
Since the area of the bottom face of Tank A is
Since the area of the bottom face of Tank B is
Since the combined volume is
Thus, the depth is 3.84 cm.
Suppose that the volumes of water in Tank C and Tank D are equal
The volume of Tank C is
Since Tank C starts full, it drains at
Since Tank D starts empty and fills at
Since these volumes are equal, then
Suppose that the depth of water in Tank D at this instant is
This is because, as Tank D fills, the ratio of the side length of the base to the height will remain constant. This ratio for the whole tank is
The volume of a square-based pyramid whose base has an edge length of
Therefore,
Therefore, at the instant when the volume of water in Tank C equals the volume of water in Tank D, the depth of water in Tank D is 5 cm.
For example, if
To see this, we note first that
For
Thus the result of the product
Since the only values for
To be a multiple of 2000,
Suppose that
In this case,
Alternatively, we could note that
Therefore,
We note first that
For
Assume that
We note that we need to count possible values of
As a result, we may assume that
Among the possible values of
Therefore, each of
Suppose
Suppose
Suppose
In total, there are
Since
A positive integer is divisible by 5 exactly when its units digit is either 0 or 5. Therefore,
A positive integer is divisible by 4 exactly when the integer formed by its last two digits is divisible by 4. Since 2 is not divisible by 4,
Since
none of
none of
exactly one of
In other words, exactly one of
For the moment, we suppose that
Suppose that
Since
Since
Thus,
Suppose that
Since
We write
Thus,
Since the final two digits of
Since 60 is divisible by 4, then
If the final two digits of
Since
We now need to determine what, if any, products
Suppose that
Suppose that
Suppose that
Suppose that
Finally, if
Therefore, we could have
Finally, we arrange
In the first case, there are 4 spots into which one can place the 2, and so there are 4 ordered quadruples in this case.
In the second case, there are 4 spots into which to first put the 6 and then 3 remaining spots into which to put the 7, leaving the 9s to be put without choice into the remaining spots. This gives
In total, there are