Wednesday, November 15, 2023
(in North America and South America)
Thursday, November 16, 2023
(outside of North American and South America)
©2023 University of Waterloo
The bus leaves at exactly 7:43 a.m., which means that after the
bus has travelled for exactly
The bus arrives at its destination at exactly 8:22 a.m., which is
exactly
Therefore, the number of minutes the trip took was
Answer:
Given that
Answer:
We label the vertices of the figure by the letters
From the information given in the questions,
Because of the fact that two line segments meet at a right angle if they
meet at all, the sum of the lengths of the horizontal edges
Therefore,
By similar reasoning,
Using that
The perimeter is given to be
Answer:
Since
This means each face of the larger cube is made up of
A cube has six faces, so the surface of the cube is made up of
Of the
The remaining
There are
Observe that
The
The
The
Observe that
The goal is to minimize the sum of the integers showing on the surface
of the larger cube. This means we want to position each small cube so
that the sum of the integers on its visible faces is as small as
possible.
Each small cube has a face with the integer
The
The smallest possible total of the integers on
The faces showing
Each of the
The smallest possible sum of the integers on
The faces that have the integers
This means these
By minimizing the total for each of the
Answer:
Let the distance between
The total distance travelled by the hiker was
The total distance walked on flat ground was
Therefore, the total time the hiker spent walking on flat ground was
The total distance walked uphill was
Therefore, the total time spent walking uphill was
The total distance walked downhill was
Therefore, the total time spent walking downhill was
Since the hiker left point
Of these
The total times spent walking on flat ground, uphill, and downhill were
Therefore, we have
This simplifies to
From earlier, the total distance travelled by the hiker was
Answer:
In the diagram below, a dashed vertical line and a dashed
horizontal line has been drawn through each of
These four lines, together with the
We will use the convention that a point on the horizontal boundary
between two regions is considered to be in the top region, and a point
on the vertical boundary between two regions is considered to be in the
rightmost region.
This means, for example, that point
There are three other regions between the
For each of these six regions, we will examine the values of
First, we make some general observations about how
The length of the shortest path from
Suppose the positive difference between the
Then there must be at least
However, a path of length
For the example in the problem statement, the coordinates of
Similarly, the length of the shortest path from
Region 1:
If
The length of the shortest path from
Adding these lengths, we get that
For the point
For the point
Continuing in this way, if
Therefore, if
Region 2:
If
The length of the shortest path from
The length of the shortest path from
In general, when
For both of the points
For the points
By reasoning similar to that which was used when examining Region
1, we see that for every even integer
Region 6:
If
Using similar calculations to the previous cases, one can check that
Similar to the observations for Regions 1 and
2, the smallest possible value of
As well, if
Therefore, for every even positive integer
Finally, we will examine Region 3. We will not need to carefully examine Region 4 or Region 5 to answer the question. This will be explained after Region 3 is examined.
Region 3:
If
In this case, it can be checked that
The smallest possible value of
If exactly one of
Thus, for
To find points
This corresponds to the
Thus, there are
Continuing in this way, if
of which there are exactly
We now suppose
There are only
Regions 1, 2, and 6
contain at most
Therefore, Region 3 must contain at least
From above, we have that for each positive integer
We need
It is easily checked that no points
Moreover, since
Therefore, there are exactly
This means
Answer:
Since
Since
The shaded region is the region inside
The area of
The area of
Since
The area of the shaded region is given to be
The area of
The area of
Since
The area of
The given condition means that
Taking square roots of both sides and noting that
Multiplying both sides of this equation by
Dividing both sides of this equation by
The RD sum of
The integer with digits
The integer with digits
The RD sum of
Suppose the RD sum of
From part (b), we have that
If
Therefore,
Since
This means
If
Substituting this into
We also know that
Therefore, it is not possible that
Using that
Since
Since
There are
Suppose that
From part (b), this means
Suppose there is some other integer,
Again by part (b), this means
Therefore, we have
For convenience, set
The equation above simplifies to
Rearranging this equation, we get
After factoring, we have
Observe that
The quantities
Moreover, since
The integer
By similar reasoning,
Now recall that each of
This means that each of
The difference between two integers that are at least
We have shown that
The only such integer is
Since
Therefore,
We have shown that if
In other words, if
This means, to count the number of four-digit RD sums, we can count the
number of pairs
Thus, the answer to the question is the number of pairs
If
The largest that
Therefore, if
Thus, we get
If
As with the case when
If
Continuing this reasoning, we find that we get
If
It is not difficult to see that
If
Therefore, there are no more cases to consider, and the number of
possible four-digit RD sums is
The positive multiples of
The positive multiples of
The positive multiples of
The positive multiples of
Consider an integer
Factoring, the quantity
Since
This also implies that
We also have that
Therefore,
This shows that
By factoring
Similar to the reasoning above,
This implies that
As well, since
By the given conditions on Row
Therefore, we can show that
We will now justify that for all
Consider the quantity
Note that any two multiples of
We are assuming that
Since
The integer
We will show that
When
Since
Therefore,
In order to show that
For the remainder of the solution, we will assume that
Note that if
Since
Since
This shows that
The previous two sentences combine to show that
This means we have
For each integer
Suppose
The expression
After some trial and error, you might notice that
We are assuming
The difference between two multiples of
If
We have now shown that if
This kind of reasoning can be applied to the remaining eight
cases.
For example, suppose
Observe that
If
If
We will now quickly include the results of examining the remaining
seven cases. The calculations are similar to those for Rows
To recap, we assumed that
Therefore, if