May 2016
© 2016 University of Waterloo
Evaluating,
Answer: 4
Since 20 minutes is
Since 30 minutes is
In total, Zeljko travelled
Answer: 20
Using the definition,
Answer: 79
We call the numbers tossed on the two dice
There are 6 possible values for each of
Therefore, there are
If
If
If
If
If
If
Thus,
Answer:
An integer is divisible by 15 exactly when it is divisble by both 5
and 3.
For a positive integer to be divisible by 5, its units (ones) digit
must be 5 or 0.
For a positive integer to be a palindrome, its first and last digits
must be the same.
Since a positive integer cannot have a first digit of 0, then the
first and last digits of a palindrome that is divsible by 5 must be
5.
Therefore, a five-digit palindrome that is divisible by 15 (and hence
by 5) must be of the form
For
This gives
The next largest five-digit palindrome of the form
Therefore, the largest five-digit palindrome that is divisible by 15
is
Answer: 59895
Since this arithmetic sequence of integers includes both even and odd
integers, then the common difference in the sequence must be odd. (If
it were even, then all of the terms in the sequence would be even or
all would be odd.)
Suppose that this common difference is
Note that 555 is one of the terms in the sequence.
If
If
If
We can also rule out the possibility that
Since there are 14 terms in the sequence and the common difference is
, then the largest term is larger than the smallest term.
Since all terms are between 500 and 599, inclusive, then the maximum possible difference between two terms is 99.
This means that. Since is an odd integer, then cannot be 9 or greater.
Since
Can we actually construct a sequence with the desired properties?
Starting with 555 and adding and subtracting 7s, we get the sequence
(Note that this sequence cannot be extended in either direction and
stay in the desired range.)
Therefore, the smallest of the 14 house numbers must be 506.
Answer: 506
Since
Therefore,
We want to find the coordinates of points
Since
Since
Note that
Since the slope of
The equation of the line through
An arbitrary point on this line can be written in the form
The points
Thus, given an arbitrary point
The slope of
We have the following equivalent equations:
Substituting into
We can check that the points
Thus, the coordinates of the points
Answer:
Suppose that Claudine has
When Claudine divides all of her candies equally among 7 friends, she
has 4 candies left over. This means that
When Claudine divides all of her candies equally among 11 friends, she
has 1 candy left over. This means that
Note then that
In other words,
This means that
Since Claude has
Putting these pieces together, we want to find the smallest
non-negative integer
To do this efficiently, we list, in increasing order, the positive
integers that are 10 less than a multiple of 77 until we reach one
that is a multiple of 19:
This means that the smallest possible value of
Answer: 40
By symmetry, the area of the shaded region on each of the four
trapezoidal faces will be the same. Therefore, the total shaded area
will be 4 times the shaded area on trapezoid
Let
Let
Since
Therefore, the area of trapezoid
The shaded area on trapezoid
Since
This means that
Since
Since the sum of the heights of these triangles is the height of the
trapezoid, then the height of
Thus, the area of
Therefore, the shaded area on trapezoid
To complete our solution, we need to calculate
To do this, we begin by dropping perpendiculars from
Since
Thus,
Since
Since
Now
Finally, we have
By the Pythagorean Theorem,
This means that the total shaded area is
Answer:
The inequality
Since the square of any real number is at least 0, then this
inequality is equivalent to
Similarly, since
Thus, the original system of inequalities is equivalent to the system
of equations
Factoring, we obtain
If
Since
In this case, we get the pairs
If
Since
In this case, we get the pairs
Therefore, the pairs that satisfy the original system of inequalities
are
Answer:
Evaluating,
Answer:
Since the average of 5 numbers is 7, then the sum of the 5 numbers is
Therefore,
Answer: 13
Since
Answer:
Starting with ABC, the magician
swaps the first and second cups to obtain BAC, then
swaps the second and third cups to obtain BCA, then
swaps the first and third cups to obtain ACB.
The net effect of these 3 moves is to swap the second and third
cups.
When the magician goes through this sequence of moves 9 times, the net
effect starting from ABC is that he swaps the second and third cups 9
times.
Since 9 is odd, then the final order is ACB.
Answer: ACB
Since
Thus,
Answer: 2
Since
Therefore,
Alternatively, we can factor the left side of the equation
Since
Now,
Since
Answer: 185
The number of two-digit integers in the range 10 to 99 inclusive is
We count the number of two-digit positive integers whose tens digit is
a multiple of the units (ones) digit.
If the units digit is 0, there are no possible tens digits.
If the units digit is 1, the tens digit can be any digit from 1 to 9.
This gives 9 such numbers.
If the units digit is 2, the tens digit can be
If the units digit is 3, the tens digit can be
If the units digit is 4, the tens digit can be
If the units digit is any of
In total, there are
Answer:
The area of pentagon
Since the area of rectangle
The area of
Since
Thus, the area of
Therefore, the area of pentagon
Answer: 1440
The integer with digits
Similarly, the integer with digits
From the given addition, we obtain
Dividing both sides by 3, we obtain
Rearranging gives
Since
Since
If
If
(Neither
Therefore,
(Checking, we see that
Thus,
Answer: 57
Suppose that, on the way from Appsley to Bancroft, Clara rides
Then on the reverse trip, she travels
Clara rides downhill at 24 km/h, on level road at 16 km/h, and uphill
at 12 km/h.
Since the original trip takes 2 hours, then
Since the reverse trip takes 2 hours and 15 minutes (or 2.25 hours),
then
Adding these two equations, we obtain
The distance from Appsley to Bancroft is
Answer: 34
We write out the first few terms of the sequence using the given rule:
Since
Answer:
Since Austin chooses 2 to start and Joshua chooses 3 next, we can list all of the possible orders of choices of numbers:
Order of Numbers | Joshua’s 1st Rd Score | Austin’s 1st Rd Score | Joshua’s 2nd Rd Score | Austin’s 2nd Rd Score | Joshua’s Total | Austin’s Total |
---|---|---|---|---|---|---|
2, 3, 1, 4, 5 | 6 | 3 | 4 | 20 | 10 | 23 |
2, 3, 1, 5, 4 | 6 | 3 | 5 | 20 | 11 | 23 |
2, 3, 4, 1, 5 | 6 | 12 | 4 | 5 | 10 | 17 |
2, 3, 4, 5, 1 | 6 | 12 | 20 | 5 | 26 | 17 |
2, 3, 5, 1, 4 | 6 | 15 | 5 | 4 | 11 | 19 |
2, 3, 5, 4, 1 | 6 | 15 | 20 | 4 | 26 | 19 |
Since Austin’s total is larger than Joshua’s total in 4 of the 6
games, then the probability that Austin wins is
Answer:
Suppose that the cone has radius
Since the lateral surface area of the cone is
Thus,
Since the lateral surface area is
Now, by the Pythagorean Theorem,
Since
Thus, the volume of the cone is
Suppose that a sphere of equal volume has radius
Thus,
Answer:
Using logarithm rules,
Answer:
We want to determine the number of four-digit positive integers of the
form
A positive integer is divisible by 45 exactly when it is divisible by
both 5 and 9.
For an integer to be divisible by 5, its units digit must be 0 or
5.
Thus, such numbers are of the form
For an integer to be divisible by 9, the sum of its digits must be
divisible by 9.
Case 1:
The sum of the digits of
Since
For
If
If
There are 11 possible four-digit numbers in this case.
Case 1:
The sum of the digits of
Since
For
If
If
If
There are 12 possible four-digit numbers in this case.
Overall, there are
Answer: 23
Since
(This comes from the fact that the sum of the interior angles in a
regular
Therefore,
Also, the reflex angle at
Since the sum of the angles in quadrilateral
Answer: 18
There are
We count the number of ways in which 3 marbles of the same colour can
be chosen.
There are 0 ways of choosing 3 magenta or 3 puce marbles, since there
are fewer than 3 marbles of each of these colours.
There is 1 way of choosing 3 cyan marbles since there are exactly 3
cyan marbles in total.
Since there are 4 ecru marbles, there are
Since there are 5 aquamarine marbles, there are
Since there are 6 lavender marbles, there are
Thus, there are
Since there are
For this probability to equal
Answer: 38
We begin by comparing pairs of the given expressions:
Therefore,
when
when
when
Noting that
when
when
when
when
This tells us that
Since
Since
Since
Comparing the maximum values of
Answer:
Suppose that
Now
Also,
In
But
This means that
Finally, we obtain
Answer:
Solution 1
Let
Suppose that, in the first line, there are
Suppose that, in the second line, there are
Since there are 75 horses, then
Since there are 10 more cows opposite cows than horses opposite
horses, then
The total number of animals is
But
Now
Solution 2
Let
Let
Let
Since the total number of horses is 75, then
Also, the given information tells us that
The total number of animals is thus
Answer: 170
Suppose that the three small circles have centres
For the larger circle to be as small as possible, these three small
circles must just touch the larger circle (which has centre
Note that
Let
Since the larger circle is tangent to each of the smaller circles at
Consider the point
Since
In a similar way, we can find that
Now since the circles with centres
Similarly,
Consider
By symmetry,
Let
Since
This means that
Therefore,
Thus, the radius of the smallest circle enclosing the other three
circles is
Answer:
If we impose no restrictions initially, each of the 7 patients can be
assigned to one of 3 doctors, which means that there are
To determine the number of ways in which the patients can be assigned
to doctors so that each doctor is assigned at least one patient, we
subtract from
This can happen if all of the patients are assigned to one doctor.
There are 3 ways to do this (all patients assigned to Huey, all
patients assigned to Dewey, or all patients assigned to Louie).
This can also happen if all of the patients are assigned to two
doctors and each of these two doctors are assigned at least one
patient.
If each of the 7 patients are assigned to Huey and Dewey, for example,
then there are 2 choices for each patient, giving
Similarly, there are
Overall, this means that there are
Answer: 1806
Suppose that
Then
Therefore
Since
Therefore,
Substituting, we obtain
Answer:
Let
Then
(Note that any other point
Consider
The area of
Also, the area of
But
In other words,
So we need to determine the length of
Since
Since
Answer:
If
This means that
Since
We determine the first several terms in each sequence to see if we can
find a pattern.
By definition,
Next,
This gives
Next,
This gives
Next,
This gives
Next,
This gives
Based on this apparent pattern, we guess that, for
We will proceed based on this guess, which we will prove at the end of
this solution.
Following this guess,
We want to find all positive integers
We note that
We also know that
This means that
We note that
Similarly,
Also,
Since
We note that
Now since
Rearranging, we get
Since the left side is even, then
Since
This means that
Therefore, the possible values of
To complete our solution, we prove that
We already know that
We prove that the desired formulas are true for all
We have proven that these formulas are correct for
If we can prove that whenever the forms are correct for
So suppose that
Then
This means that
Therefore, if
This proves that
Answer: 50,51,52,53,54
(Note: Where possible, the solutions to parts (b) and (c) of each Relay
are written as if the value of
Evaluating,
The area of a triangle with base of length
Since the answer to (a) is 4, then
Since
Therefore,
Since the answer to (b) is 52, then
Answer:
Since
If
Since the answer to (a) is 9, then
We start with the given equation and complete the square:
Since the
Since the answer to (b) is 3, then
Answer:
To find the
To find the
The sum of the intercepts is
Since
Since
The area of
The area of trapezoid
The area of
The area of
Therefore,
Since the answer to (a) is 6, then
The volume of a cylinder with radius
The volume of a cylinder with radius
From the given information,
Since
Rearranging
If we assume that
Since
Under the assumption that
In particular, we obtain the pairs
Since the answer to (b) is 30, then
Answer:
Since
Since
Therefore,
Suppose that Oscar saw
From the given information, he saw
Therefore, he saw
Note that Oscar saw
Since he saw
Since the answer to (a) is 6, then
Suppose that there are
Since every row after the first contains 4 more seats than the
previous, then the remaining 19 rows contain
The numbers of seats in the 20 rows form an arithmetic sequence
with 20 terms, first term equal to
The total number of seats is equal to the sum of this series,
which equals
Since we are told that there are
Therefore,
Since the answer to (b) is 240, then
In other words, there are 82 seats in the first row.
Answer: