Wednesday, November 25, 2015
(in North America and South America)
Thursday, November 26, 2015
(outside of North American and South America)
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If
Alternatively, we could note that
Answer:
We start by considering the ones (units) column of the given sum.
From the units column, we see that the units digits of
The only digit for which this is possible is
Thus, the sum becomes
Next, we consider the tens column.
From the tens column, we see that the units digit of
The only digit for which this is possible is
Thus, the sum becomes
Next, we consider the hundreds column.
We see that the units digit of
Thus, the units digit of
The value of
Therefore,
(Putting this analysis another way, we have
We can check that
Answer:
Since the roof measures 5 m by 5 m, then its area is
When this roof receives 6 mm (or 0.006 m) of rain, the total volume of rain that the roof receives is
The rain barrel has diameter 0.5 m (and so radius 0.25 m) and a height of 1 m, and so has volume
Thus, the percentage of the barrel that is full is
Answer:
Using exponent laws, we obtain the following equivalent equations:
Therefore,
We can check by substitution that each of these values of
Answer:
Suppose that Anna guesses “cat”
When she guesses “dog”, she is correct
When she guesses “cat”, she is correct
Thus, when she guesses “cat”, she is shown
Thus, when she guesses “dog”, she is shown
(We assume that
Therefore, the total number of images of cats that she is shown is
But the number of images of cats equals the number of images of dogs.
Thus,
Therefore, the ratio of the number of times that she guessed “dog” to the the number of times that she guessed “cat” is
Answer:
Since
Now,
Thus, we have
We want to determine the number of positive integers
Re-arranging the equation, we obtain
Since
Since
Since
Now, the factorizations of
This is because a factorization
So the problem is equivalent to determining the number of positive integers
Given an integer
In order for this product to equal 6,
In other words, we want
If
If
If
If
Therefore, in summary, there are 12 such values of
Answer:
The
Since
Since
The point
Thus, the
So the coordinates of
When
Quadrilateral
Therefore, the area of
In general, quadrilateral
In terms of
Since
We can verify that if
If
Since
Thus,
We can check by substitution that each of these values of
Since
(Note that above we have multiplied both the numerator and denominator of the complicated fraction by
When
When
The following equations are equivalent for all
Therefore,
When
When
The following equations are equivalent for all
In other words, we must have
Since
Note that if
Therefore,
When
In tabular form, we have
As in (a), we start by calculating the first several values of
We will show that if
We note that this is consistent with the table shown above.
Using this fact without proof, we see that
(This list of values of
Thus, once we have proven this fact, the positive integers
Suppose that
Since
Since
Continuing in this way, we find that
To justify these statements, we note that each is true when
if
if
This tells us that, for these values of
Now
Since
Thus, the positive integers
Suppose that
Then there exist positive integers
Note that this means that the terms from
In other words, the sequence
Let
Since the terms in the sequence
From the above comment,
In other words,
Next we prove that
To do this, we first note that
Next, we consider a given term
Note that
If
Since
If
In either case, if
Therefore,
Now we consider the
Since
Since
(We have a list of
Suppose that
Now each term
Since
In particular, if
In other words, starting at each of