Wednesday, November 18, 2020
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Thursday, November 19, 2020
(outside of North American and South America)
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Between them, Markus and Katharina have
When Sanjiv gives 10 candies in total to Markus and Katharina, they now have
Since Markus and Katharina have the same number of candies, they each have
Answer: 12
Suppose that the square has side length
Each of the two rectangles thus has width
In terms of
Since the perimeter of each rectangle is 24 cm, then
Since the square has side length
Answer:
Solution 1:
Since
From
(Checking, we see that
Solution 2:
Since
From
(Checking, we see that
Answer:
We note that
Therefore,
This means that
Next, we re-write
Each of the terms
There are 86 terms in the first list and 14 terms in the second list.
Thus,
Answer:
We square the two given equations to obtain
(It is possible to solve for
Answer:
Using the second property with
Using the first property with
Using the first property with
Using the second property with
Next, we note that
Since
Since
Since
Using the second property with
Using the first property with
Here are two additional comments about this problem and its solution:
While the solution does not contain many steps, it is not easy to come up with the best steps in the best order to actually solve this problem.
There is indeed at least one function, called the Cantor function, that satisfies these properties. This function is not easy to write down, and finding the function is not necessary to answer the given question. For those interested in learning more, consider investigating ternary expansions and binary expansions of real numbers between 0 and 1, as well as something called the Cantor set.
Answer:
To determine the point of intersection, we equate the two expressions for
Therefore, the lines intersect at
To determine the point of intersection, we equate the two expressions for
Therefore, the lines intersect at
Since
The two lines have slopes
To determine the point of intersection, we equate the two expressions for
In particular, this means that the
When
Since
Therefore, the lines intersect at a point whose coordinates are integers.
To determine the point of intersection in terms of
Since
We still need to confirm that, for each of these values of
When
Therefore, when
We can verify this in each case:
Each interior angle in a regular hexagon measures
Since
Therefore, the shaded area is
To determine the area of region between the arc through
There are many ways to find the area of this triangle.
One way is to consider
Since
This means that
This means that the area of
Therefore, the area of the shaded region between the arc through
Since the region between line segment
Therefore, the total area of the shaded regions is
Let
This means that
Let
Join
By symmetry,
Let the area of one of these pieces be
Furthermore, by symmetry in the whole hexagon, each of the six shaded regions between two semi-circles is equal in area.
This means that the entire shaded region is equal to
Therefore, we need to determine the value of
Consider the region between
Since
Since
Consider
Also,
Since
Since
Therefore,
Now, we can calculate the area
The area
Since
The area of
One way to do this is to use the formula that the area of a triangle with two side lengths
Thus, the area of
This means that
Finally, the total area of the shaded regions equals
When
Therefore, the equation
Suppose first that the equation
Since
For the roots of
Therefore,
Suppose also that the equation
The roots of the equation
Furthermore, since the roots of the equation
This means that
This means that
To complete this part, we need to prove that
To do this, we use the fact that
Since
In this case,
Since any perfect square that is a multiple of 3 must be a multiple of 9 (prime factors of perfect squares occur in pairs), then
Since
Since
The goal of this solution is to show that there are infinitely many pairs of positive integers
To begin, we assume that
Thus, we write
Suppose that
These first two Assumptions are not surprising given the results of (b).
In this case, the non-zero solutions of
Since
Further, we note that since
Therefore, to find an infinite number of pairs of positive integers
Recall that
This gives
Subtracting, we obtain
We re-write this equation as
Now we suppose that
This adds a further assumption that connects the integers
Then
Under these assumptions, for
Recall that, to find an infinite number of pairs of positive integers
Setting
Because
Because
This gives
Therefore, each positive integer
Therefore, in order to complete our proof, we need to show that there are infinitely many integers
Since
Since
Thus,
Note that the positive divisors of
Suppose that
Therefore, there are infinitely many positive integers
This means that there are infinitely many pairs of positive integers
This means that there are infinitely many pairs of positive integers