Tuesday, April 4, 2023
(in North America and South America)
Wednesday, April 5, 2023
(outside of North American and South America)
©2023 University of Waterloo
Since the average of the 5 numbers
Therefore,
Solution 1
Adding the equations
Therefore, the average of
Solution 2
Since
Subtracting the second equation, we obtain
Thus,
The average of
Since the average of the three numbers
Therefore,
Since
Since
Thus,
Therefore,
Solution 1
The point with coordinates
A line with slope 3 moves 2 units to the right as it moves 6 units up.
Therefore, to move from
A line with slope
The distance between these
Solution 2
The line with slope
The
The line with slope
The
The distance between these
The line with equation
The line with equation
Since these lines are perpendicular, the product of their slopes is
We now need to find the point of intersection of the lines with
equations
Equating expressions for
Therefore,
Since
The sum of these divisors is
Suppose that the four consecutive integers that Fionn originally
wrote on the blackboard were
When Lexi erases one of these integers, the sum of the remaining three
integers is equal to one of the following:
We note that
Therefore, the original integers were 281, 282, 283, 284 and Lexi erased
From the given information, the 7 terms in the
arithmetic sequence are
The corresponding arithmetic sequences are
In 1 hour, Liang paints
Thus, in 2 hours, Liang paints
Edmundo needs to paint
In 1 hour, Edmundo paints
Since
Therefore, Edmundo paints for 80 minutes.
When converted to a fraction,
When an amount is increased by
When an amount is decreased by
When $400 is increased by
When this value is decreased by
Therefore,
The quadratic function
The discriminant of this function is
This final inequality is equivalent to
Therefore, the smallest positive integer that satisfies this inequality,
and hence for which
Using the cosine law in
Therefore,
Substituting into the previous equation,
Solution 1
We make two copies of the given triangle, labelling them
The combined area of these two triangles is
Next, we rotate
and join the two triangles together:
We note that
Therefore,
Further,
The total area of parallelogram
Thus, the shaded area of
Since this shaded area is equally divided between the two halves of the
parallelogram, then the combined area of the shaded regions of
Solution 2
We label the points where the horizontal lines touch
We use the notation
Let
Thus,
Suppose that the height of
Since the height of each of the 11 regions is equal in height, then the
height of
When two triangles are similar, their heights are in the same ratio as
their side lengths:
To see this, suppose that
is similar to and that altitudes are drawn from and to and .
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Since
, then is similar to (equal angle; right angle), which means that . In other words, the ratio of sides is equal to the ratio of heights.
Since the height of
Therefore,
Similarly, since the height of
Therefore,
This result continues for each of the triangles.
Therefore,
Solution 1
We label five additional points in the diagram:
Since
Since
We are told that
Thus,
Thus,
Since
This tells us that
Therefore,
Solution 2
We add coordinates to the diagram as shown:
We determine the side length of square
The slope of the line through
This equation of this line can be written as
The slope of the line through
The equation of this line can be written as
The slope of the line through
The equation of this line can be written as
Point
Equating expressions for
Since
Point
Equating expressions for
Since
Therefore,
Each possible order in which Akshan removes the marbles
corresponds to a sequence of 9 colours, 3 of which are red and 6 of
which are blue.
We write these as sequences of 3 R’s and 6 B’s.
Since are told that the first marble is red and the third is blue, we
would like to consider all sequences of the form
There are
Of these 21 ways, some have the last two marbles being blue.
These correspond to the sequences of the form
There are
Therefore, 10 of the 21 possible sequences end in two B’s, and so the
probability that the last two marbles removed are blue is
Factoring the first equation, we obtain
To find the divisors of 2023, we first find its prime factorization:
This means that the divisor pairs of 2023 are
Since
Since
When
When
When
In total, there are
Since
Since
We note that
Since
The smallest even perfect square in this range is
Since
Therefore, the number of even perfect squares in this range is
Thus, there are 222 positive integers
Let
Using logarithm laws,
When
When
When
When
Therefore,
(It seems surprising that the solution to this equation is actually an
interval of values, rather than a finite number of specific
values.)
If there are 5 or more people seated around a table with 8
chairs, then there are at most 3 empty chairs. But there must be an
empty chair between each pair of people, and this is not possible with 5
people and 3 empty chairs.
Therefore, there are at most 4 people seated.
If there were only 2 people seated, then there would be 6 empty chairs
which would mean that at least one of the two “gaps” around the circular
table had at least 3 empty chairs, and so another person could be
seated, meaning that the table wasn’t full.
Therefore, there are at least 3 people seated.
This means that a full table with 8 chairs has either 3 or 4
people.
If there are 4 people, there are 4 empty chairs, and so there is exactly
1 empty chair between each pair of people.
Thus, people are seated in chairs
If there are 3 people, there are 5 empty chairs.
With 3 people, there are 3 gaps totalling 5 chairs, and each gap has at
most 2 chairs in it.
Therefore, the gaps must be 1, 2, 2 in some order. This is the only list
of three positive integers, each equal to 1 or 2, that adds to 5.
The gap of 1 can be between any pair of seats. In other words, the gap
of 1 could be between
Thus, with 3 people, they are seated in chairs
Suppose that
Suppose that
When
Therefore, there are between
This means that the total number of chairs is between
In other words,
Since
We note that it is possible to seat
Since
We note that it is possible to seat
We now know that, if there are
To confirm that every such value of
From the work above, we know that
The total number of gaps is
Finally, the total number of chairs is
This shows that every
Therefore, the possible values of
There are
Solution 1
For each integer
We can check that
In the problem, we are told that
This gives us the following table:
3 | 3 |
4 | 2 |
5 | 5 |
6 | 5 |
7 | ? |
8 | 10 |
Based on this information, we make the guess that for every integer
For example, this would mean that
Based on this recurrence relation (which we have yet to prove), we
deduce the values of
3 | 3 |
4 | 2 |
5 | 5 |
6 | 5 |
7 | 7 |
8 | 10 |
9 | 12 |
10 | 17 |
11 | 22 |
12 | 29 |
13 | 39 |
14 | 51 |
15 | 68 |
16 | 90 |
17 | 119 |
18 | 158 |
19 | 209 |
We now need to prove that the equation
We think about each full table as a string of 0s and 1s, with 1
representing a chair that is occupied and 0 representing an empty
chair.
Let
Let
Let
Since every full table must be in one of these categories, then
A full table with
Since there cannot be more than two consecutive 0s, we can further
specify this, namely to say that a full table with
Consider the full tables starting with 1010. Note that such strings
end with 0 since the table is circular. Removing the 10 from positions 1
and 2 creates strings of length
Consider the full tables starting with 1001. Note that such a string
ends with 0 since the table is circular. Removing the 100 from positions
1, 2 and 3 creates strings of length
Consider the full tables starting with 0100. Removing the 100 from
positions 2, 3 and 4 creates strings of length
Consider the full tables starting with 0101. Removing the 01 from
positions 3 and 4 creates strings of length
Consider the full tables starting with 0010. These strings must begin
with either 00100 or 00101.
If strings start 00100, then they start 001001 and so we remove the 001
in positions 4, 5 and 6 and obtain strings of length
If strings start 00101, we remove the 01 in positions 4 and 5 and obtain
strings of length
These 6 cases and subcases count all strings counted by
Therefore,
Solution 2
Extending our approach from (b), the number of people seated at a
full table with 19 chairs is at least
Since the number of people is an integer, there must be 7, 8 or 9 people
at the table, which means that the number of empty chairs is 12, 11 or
10, respectively.
Suppose that there are 9 people and 9 gaps with a total of 10 empty
chairs.
In this case, there is 1 gap with 2 empty chairs and 8 gaps with 1 empty
chair.
There are 19 pairs of chairs in which we can put 2 people with a gap of
2 in between:
Once we choose one of these pairs, the seat choice for the remaining 8
people is completely determined by placing people in every other
chair.
Therefore, there are 19 different full tables with 9 people.
Suppose that there are 8 people and 8 gaps with a total of 11 empty
chairs.
In this case, there are 3 gaps with 2 empty chairs and 5 gaps with 1
empty chair.
There are 7 different circular orderings in which these 8 gaps can be
arranged:
If the three gaps of length 2 are consecutive, there is only one
configuration (22211111).
If there are exactly 2 consecutive gaps of length 2, there are 4
relative places in which the third gap of length 2 can be placed.
If there are no consecutive gaps of length 2, these gaps can either be
separated by 1 gap each (21212111) with 3 gaps on the far side, or can
be separated by 1 gap, 2 gaps, and 2 gaps (21211211). There is only one
configuration for the gaps in this last situation.
There are 7 different circular orderings for these 8 gaps.
Each of these 7 different orderings can be placed around the circle of
19 chairs in 19 different ways, because each can be started in 19
different places. Because 19 is prime, none of these orderings
overlap.
Therefore, there are
Suppose that there are 7 people and 7 gaps with a total of 12 empty
chairs.
In this case, there are 2 gaps with 1 empty chair and 5 gaps with 2
empty chairs.
The 2 gaps with 1 empty chair can be separated by 0 gaps with 2 empty
chairs, 1 gap with 2 empty chairs, or 2 gaps with 2 empty chairs.
Because the chairs are around a circle, if there were 3, 4 or 5 gaps
with 2 empty chairs between them, there would be 2, 1 or 0 gaps going
the other way around the circle.
This means that there are 3 different configurations for the gaps.
Each of these configurations can be placed in 19 different ways around
the circle of chairs.
Therefore, there are
In total, there are
Solution 3
As in Solution 2, there must be 7, 8 or 9 people in chairs, and so
there are 7, 8 or 9 gaps.
If there are 7 gaps, there are 2 gaps of 1 chair and 5 gaps of 2
chairs.
If there are 8 gaps, there are 5 gaps of 1 chair and 3 gaps of 2
chairs.
If there are 9 gaps, there are 8 gaps of 1 chair and 1 gap of 2
chairs.
We consider three mutually exclusive cases: (i) there is a person in
chair 1 and not in chair 2, (ii) there is a person in chair 2 and not in
chair 1, and (iii) there is nobody in chair 1 or in chair 2. Every full
table fits into exactly one of these three cases.
Case (i): there is a person in chair 1 and not in chair 2
We use the person in chair 1 to “anchor” the arrangement, by starting
at chair 1 and arranging the gaps (and thus the full chairs) clockwise
around the table from chair 1.
If there are 7 gaps, we need to choose 2 of them to be of length 1, and
so there are
If there are 8 gaps, we need to choose 3 of them to be of length 2, and
so there are
If there are 9 gaps, we need to choose 1 of them to be of length 2, and
so there are
In this case, there are a total of
Case (ii): there is a person in chair 2 and not in chair 1
We use the same reasoning starting with the person in chair 2 as the
anchor.
Again, there are 86 full tables in this case.
Case (iii): there is nobody in chair 1 or chair 2
Since there is nobody in chair 1 or chair 2, there must be a person
in chair 3 and also in chair 19, which fixes one gap of 2 chairs.
Here, we use the person in chair 3 as the anchor.
If there are 7 gaps, there are 2 gaps of 1 chair and 4 gaps of 2 chairs
left to place. There are
If there are 8 gaps, there are 5 gaps of 1 chair and 2 gaps of 2 chairs
left to place. There are
If there are 9 gaps, there are 8 gaps of 1 chair and 0 gaps of 2 chairs
left to place. There is 1 way to do this.
In this case, there are a total of
In total, there are
Since
Since
These fractions can continue to be grouped in groups of 3 with the
last full group of 3 satisfying
The last term is
If the given sum is
For every positive integer
Every positive integer
We can thus write
In other words, we want to find a polynomial
We will show that the polynomial
If
This means that
Therefore,
If
This means that
Therefore,
If
Since
Therefore,
This means that the polynomial
Before working on the specific question we have been asked, we
simplify the given expression for
This means that if
This means that, for a fixed positive integer
Thus,
We will assume that
We note that when
Next, suppose that
Since
In this case, consider
We can write this as
We know that
Thus,
This means that the
To show that
Consider
Since
Consider
Since
Consider the term
We have
Thus,
Also,
Therefore, if
Here is an alternative approach so show that
As above, we look for at least 3 integers
The positive integer
This pair of inequalities is equivalent to the pair of inequalities
The following three pairs
This shows that