May 2015
© 2015 University of Waterloo
The inequality
Since
Therefore, the smallest positive integer
Answer: 99
We label the diagram as shown.
Since
Answer: 21
Let
Then
Since 3 of the 20 friends do not like to ski or snowboard, then 17
like to either ski or snowboard or both.
Thus,
Therefore, 7 of the friends like to both ski and snowboard.
Answer: 7
Since
Since
Since
Therefore,
Answer: 3
We note that
For
Therefore,
We try the smallest two-digit possibilities for
If
If
If
Therefore, the smallest two-digit positive integer
Answer: 20
Suppose that the distance that Clara has to travel is
When she travels at an average speed of 20 km/h, she arrives half an
hour early.
Thus, she travels for
When she travels at an average speed of 12 km/h, she arrives half an
hour late.
Thus, she travels for
Adding these two equations, we obtain
Thus,
This means that if Clara travels at an average speed of 15 km/h, then
it will take her exactly
Answer: 15
Since the sum of any four consecutive terms is 17, then
Also,
Similarly,
But
Answer: 7
Suppose that the prism has
Since the sum of all of the edge lengths is 24, then
(The prism has 4 edges of each length.)
Since the surface area is 11, then
(The prism has 2 faces that are
By the Pythagorean Theorem,
Again, by the Pythagorean Theorem,
But
Therefore,
Finally,
Answer: 5
Since the parabola opens upwards and has only one
Since the point
We will obtain a second equation for
Since
Since
To get from
Thus, to get from
This means that the coordinates of
To see why this “over and up" approach is correct, draw a vertical
line from
Since
Note that
Since
But
This means that
Since
Thus, the coordinates of
Since
Dividing this equation by the first equation
Since
Thus,
Since
Therefore, the equation of the parabola is
Answer:
First, we note that
Next, we note that
This is because since
Since
In particular,
Since multiples of 257 differ by 257, then the largest multiple of 257
less than
The next multiples of
Those between
The sum of these values is
Answer: 2054
Re-arranging the terms and grouping into pairs,
Answer: 100
When the edge lengths of 10 are increased by 10%, the new edge lengths
are 11.
Therefore, the volume of the resulting cube is
Answer: 1331
Suppose that the cylinder has radius
Since the circumference of its circular faces is
Therefore, the volume of the cylinder is
Answer:
Every prime number other than 2 is odd.
Since
Thus, at least one of
Since
Since
Since
Therefore, there are two triples
Answer: 2
Since
Since
Since the sum of the angles in
Now
Thus,
But
Thus,
Since
Therefore,
Answer:
Note that
Thus,
This product is divisible by
Since
Therefore, there are
Answer: 4
Since
Thus,
Answer: 3
To find the largest possible three-digit positive integer with no
repeated digits, we put the largest possible digit in each position,
starting with the hundreds digit (9), then the tens digit (8), and
finally the ones (units) digit (7).
Similarly, to find the smallest possible three-digit positive integer
with no repeated digits, we put the smallest possible digit in each
position, starting with the hundreds digit (1 because the leading
digit cannot be 0), then the tens digit (0), and finally the ones
(units) digit (2).
The difference between the numbers
Answer: 885
Since
Since
Since
Set
Then
Since
Since
When
When
When
When
When
Therefore, there are 5 pairs that satisfy the given conditions:
Answer: 5
Since
Thus,
But
Therefore,
Answer:
Since
Therefore,
Answer: 0
For a positive integer to be a perfect square, its prime factors must
occur in pairs.
Since only one number from the list 1, 2, 3, 4, 5, 6, 7, 8, 9 is
divisible by 5 (namely 5) and only one number from the list is
divisible by 7, then neither 5 nor 7 can be one of the three integers
chosen.
Therefore, we need to choose three numbers from the list 1, 2, 3, 4,
6, 8, 9.
We note that
Are there perfect squares larger than
If the product of three integers is odd, then all three integers are
odd. The only three integers in the list that are odd are 1, 3, 9.
Their product is 27, which is not a perfect square.
Therefore, any perfect square larger than 144 that can be made must be
even.
Further, note that the largest possible product of three numbers in
the second list is
The even perfect squares between 144 and 432 are
We cannot make 196 or 400, since no number in the list is a multiple
of 7 or 5.
We cannot obtain a product of
We cannot obtain a product of
Therefore, 144 is the largest perfect square which is the product of
three different one-digit positive integers.
Answer: 144
Suppose that the distance from Point
From the given information,
The amount of the time, in seconds, that it takes Mario to walk along
the sidewalk when it is moving is
Here,
Answer: 30 seconds
If a square has side length
Since we are told that the length of the diagonal is
Thus,
The area of the square is
Answer:
In the diagram, every path that starts at the top and follows to the
bottom spells the word “WATERLOO".
Therefore, we count the total number of paths from the top to the
bottom.
There is one path leading from the W to each of the A’s.
The number of paths leading to a given letter below the A’s is the sum
of the numbers of paths leading to the two letters above that point to
the given letter.
This is because each path leading to a given letter must go through
exactly one of the letters above pointing to the given letter.
We calculate the number of paths leading to each letter in the grid by
proceeding downwards through the grid:
Therefore, there are 20 paths that spell “WATERLOO".
Answer: 20
Using exponent and trigonometric laws,
The smallest positive angle for which one of these is true is
Answer:
Using exponent laws,
Answer: 5625
For the quadratic equation
This is because if the roots are
Then
We are told that the roots of
Then
Therefore,
(We could also solve the problem by directly calculating the roots of
Answer: -
First, we note that
This means that there are 220 different combinations of 3 eggs that
can be chosen from 12 eggs.
Zach randomly gives 3 eggs to each of the 4 children.
The probability that a specific one of these 4 children receives the
specific set of all 3 special eggs is
The probability that any one of these 4 children receives a specific
set of 3 eggs is
Therefore, the probability that only one child receives an egg that
contains a prize is
Answer:
If
Therefore,
Answer: 2015
We first note that the coordinates of
Since
Since
Since
Let
Then
Since
We note that
Thus, the possible perfect square values of
Therefore, there are 6 integers
Answer: 6
Solution 1
Let
Extend
Then
By the Pythagorean Theorem in
By the Secant-Secant Theorem,
From the first equation,
From the second equation,
Substituting, we obtain
Since
Also,
Solution 2
Let
Join
Since
By the Pythagorean Theorem,
Therefore,
By the cosine law in
Answer:
Suppose that the largest integer in
We are told that
Thus,
Since
That is,
From this,
Now, the sum of the elements of
But all of the elements of
Therefore,
If
This means that there are two elements in addition to
But each of these elements must be at least
This is a contradiction, so there cannot be 4 elements in
If
This means that there are three elements in addition to
If
This is possible with elements 17, 18, 19.
In this case,
Therefore, the minimum possible value of
Answer: 5
We cut the lateral surface of the cone along line segment
This forms a sector of a circle with radius
We label the second endpoint of the arc of this sector as
Since the circumference of the base of the original cone is
Since a full circle with radius 4 has circumference
Point
The shortest path starting at
Since
The point
We can then calculate the area of
Therefore, the length of
(We could also calculate the length
Answer:
We label the regions with unknown integers
The given list of integers includes 4 odd integers (
We proceed by focusing on
In this case, none of
Therefore, all four odd integers have to be used for
Note that 3 and 9 cannot be adjacent, by Rule 2.
Since 3, 5 and 9 are all used among
Therefore,
There are 4 choices for
For each of these, there are 4 choices of placement for 3.
Once 3 is placed, the position of 9 is determined (it must be
“opposite" 3).
Once 3 and 9 are placed, there are 2 choices of position for 5 and
then 1 choice for 1.
In this case, there are
Here, there are 2 choices for
The other possible value for
We cannot use 3 or 4 even numbers among
Therefore, at most 2 even numbers can be used.
But there are only 2 odd numbers left (1 and 5), so exactly two
even numbers must be used and these cannot be adjacent. The even
numbers used cannot include 6, 10, 12, by Rules 2 and 3 (each even
number used will be adjacent to
Therefore, only 2, 4, 8, 14 can be used.
There are thus 2 choices for
In this case, there are
Since 2 even numbers cannot be adjacent among
Suppose that there are 3 odd numbers among
Note that 3 and 9 cannot be adjacent, so must be opposite.
There are 4 choices for placement of 3, which gives 1 placement
for 9 (opposite 3), and 2 choices for 1 (in either open spot).
The remaining integer will be even and cannot be 6, 10 or 12, and
so there are 4 choices for the remaining spot.
In this subcase, there are
Suppose that there are now 2 odd numbers among
There are 4 choices of placement for the smaller of these numbers
and then the larger one is placed opposite.
The even numbers are chosen from among
In this subcase, there are thus
In this case, there are thus
Since 2 even numbers cannot be adjacent among
Suppose that there are 3 odd numbers among
Note that 3 and 9 cannot be adjacent, so must be opposite.
There are 4 choices for placement of 3, which gives 1 placement
for 9 (opposite 3), and 2 choices for 5 (in either open spot).
The remaining integer will be even and cannot be 6 or 12, and so
there are 5 choices for the remaining spot. (Note that the even
number is now opposite 5 so could equal 10.)
In this subcase, there are
Suppose that there are now 2 odd numbers among
If 3 and 9 are used, there are 4 choices of placement for 3 and
then 1 choice of placement for 9.
The even numbers are chosen from among
In this subcase, there are thus
If 3 and 5 are used, there are 4 choices of placement for 3 and
then 1 choice of placement for 5.
The even numbers are chosen from among
In this subcase, there are thus
If 9 and 5 are used, there are again
In this case, there are thus
Therefore, in total there are
Answer: 520
(Note: Where possible, the solutions to parts (b) and (c) of each Relay
are written as if the value of
Evaluating,
The average of the numbers in the first list is
The average of the numbers in the second list is
Therefore,
Since the answer to (a) is 8, then
Since the two lines intersect at
Therefore,
Since
Since the answer to (b) is 1, then
Answer:
Since
If
Since the answer to (a) is 6, then
We use the Pythagorean Theorem repeatedly and obtain
Since
Answer:
Since
We treat
The height of
Therefore, the area of
According to the given information, at last night’s rehearsal,
Canada’s Totally Musical Choir (CTMC) spent
As a percentage of the full 75 minute rehearsal, this is
Using the rule that the sum of two entries in consecutive boxes in
one row is the entry between these boxes in the row above, the
entries in the second row from the bottom should be
Thus, the first entry in the third row from the bottom should be
Also, the two entries in the second row from the top should be
Now the given entry
Thus,
Also, the top entry
Thus,
Since
Since
Thus,
Since the answer to (b) is 32, then
Answer:
A rectangular prism has six faces.
Since the prism has edge lengths 2, 3 and 4, then the prism has
two faces with dimensions
Therefore, the surface area of the prism is
Since
Also,
In
Thus,
Since the answer to (a) is 52, then
We do some preliminary work assuming that the number
Since
Since
Since
Next, consider
Since
Next consider
Since
Next consider
Since
Since the answer to (b) is 56, then
Since
Furthermore,
Therefore, there are 5 integers
Answer: