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 each box contains 12 trays, then 6 boxes contain
Since Paul has 4 extra trays, then he has
Since each tray can hold 8 apples, then the trays can hold
Answer: 608 apples
Since the rabbit runs 3 times as quickly as the skunk, then the rabbit takes
Since the rabbit runs 5 times as quickly as the turtle, then the turtle takes 5 times as long as the rabbit, or 10 minutes, to finish the race.
Answer: 10 minutes
Solution 1
With 6 crayons in the box, there are
Using similar reasoning, with 3 red crayons in the box, there are
Therefore, the probability that Jakob removes 2 red crayons is
Solution 2
We label the crayons in the box R1, R2, R3, B1, B2, G1.
The possible pairs that Jakob can choose are
R1R2, R1R3, R1B1, R1B2, R1G1, R2R3, R2B1, R2B2, R2G1,
R3B1, R3B2, R3G1, B1B2, B1G1, B2G1
for a total of 15 pairs.
Of these 15 pairs, three pairs (R1R2, R1R3, R2R3) consist of two red crayons.
Therefore, the probability that Jakob removes 2 red crayons is
Solution 3
We remove the two crayons one after the other. In order for both crayons to be red, both the first and second crayons removed must be red.
When the first crayon is removed, the probability that it is red is
After 1 red crayon is removed, there are 5 crayons remaining, 2 of which are red.
Therefore, the probability that the second crayon removed is red is
Thus, the probability that both crayons are red is
Answer:
Since
Therefore,
Since
Thus,
Therefore, the sum of the digits of the integer
Since the sum of the digits of
Answer:
When we translate the parabola and hexagon horizontally, the distance between the
This means that we can position the hexagon and parabola in a horizontal position that is more convenient.
Thus, we position the hexagon and parabola so that the
Since
In other words, the coordinates of
Next, we drop a perpendicular from
Since
Thus,
Since
Since the coordinates of
Similarly,
Since the hexagon is symmetric across the
Suppose that the parabola has equation
Since the points
Substituting into the second and third equations, we obtain
Substituting
Therefore, the equation of the parabola is
We find the
Thus, the parabola crosses the
The distance between these two points is
Answer:
Solution 1
When
For all real numbers
Rearranging, we obtain
Setting
Thus,
Setting
Thus,
Since
For equality to occur in this inequality, we need equality to occur in both of the component inequalities. (If one inequality is actually “
We note that
We now need to determine the value of
Since
When
Since
When
This gives
Since
Finally, this gives
Solution 2
Let
The given equation becomes successively:
Therefore,
From the first equation,
Since
Since
We can then proceed as in Solution 1 to obtain
Answer:
To find the
Therefore, the
Solution 1
To find the equation of the line that is symmetric across the
Solution 2
The line with equation
Since the letter A is symmetric about the
The equation of this line is thus of the form
Since it passes through the point
Thus, the equation of this line is
Since the
Since both lines pass through
Therefore, the area of the triangle is
Solution 1
We note first that
Since the top vertex of the shaded triangle lies at
The bottom right vertex of the shaded triangle lies at the point of intersection of the line with equation
At this point,
By symmetry, the
Therefore, the base of the shaded triangle has length
Since the area of the shaded triangle is
Putting all of this together and using the area obtain in (c), we have
Solution 2
Since the base of the shaded triangle is parallel to the base of the larger triangle, then their base angles are equal and so the triangles are similar.
Since the area of the shaded triangle is
Furthermore, the height of the shaded triangle is
Since the height of the larger triangle is 6, then the height of the shaded triangle is
Since the top vertex of the shaded triangle lies at
Thus,
Re-arranging and noting that
Factoring, we obtain the following equivalent equations:
Since
Since
This means that
There are two possibilities:
Therefore, the pairs of positive integers that solve the equation are
Since
Since
Since
Since
Since
We make a table to determine the possible values:
Since
Since
Since
Since
Since
We consider the possibilities that
In these cases,
These give the pairs
Since
Therefore, there are at least two pairs
If
If they do not overlap, then such a string must be
A 4 character string including the substring
Solution 1
Since there are 3 choices for each character in such a string, then there are
We count the number of such strings that do not include the substring
Let
Let
Let
Let
Note that
Also,
Suppose that
Consider a string of length
Consider a string of length
Consider a string of length
Therefore,
First, we calculate
Every string counted by
Starting from the left, this first occurrence can begin in any of the positions
Suppose that
We determine the number of strings counted by
In such a string, positions
This leaves
Based on the restrictions for the string, there are no restrictions for the characters in each of the
Therefore, there are 3 choices for each of these positions and so
For the
Since position
If position
If position
In general, if we consider the character at position
if position
if position
if position
In other words, for each position from
Note that this formula is valid when
Therefore, there are
Adding from
Since