Tuesday, April 12, 2022
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Wednesday, April 13, 2022
(outside of North American and South America)
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Regular hexagon
Since
The radius of the circle is equal to
Since
Since
Using the Pythagorean Theorem in right-angled
Alternatively, notice that
The diagonals
Thus the area of
The area of the shaded region is determined by subtracting the
area of
Thus, the area of the shaded region is
Since
With 1 kg of muffin batter, exactly 24 mini muffins and 2 large
muffins can be made. Thus with 2 kg of muffin batter, exactly
Solution 1
With 2 kg of muffin batter, exactly 36 mini muffins and 6 large
muffins can be made. With 2 kg of muffin batter, exactly 48 mini muffins
and 4 large muffins can be made. Adding these, we get that with
Solution 2
Let
Since 2 kg of muffin batter makes exactly 36 mini muffins and 6 large
muffins, then
Subtracting the second equation from 3 times the first equation, we get
Substituting
Since
Since
Thus, exactly 84 mini muffins and 10 large muffins can be made with
In part (b) Solution 2, it was determined that
Therefore, the number of kilograms of muffin batter needed to make 1
large muffin is 6 times the number of kilograms of batter needed to make
1 mini muffin
In other words, the batter needed to make 1 large muffin can make 6 mini
muffins.
Then, using the amount of batter that makes 7 large muffins,
If the first number in a sequence is 3 and the sequence is
generated by the function
Let the first and second numbers in the sequence generated by the
function
Then, the first three numbers in the sequence are
Since the third number in the sequence is 7, then
Solving this equation, we get
If the second number in the sequence is 0, then
The discriminant of this equation is
Thus, there is no first number in this sequence for which the second
number is 0.
If the second number in the sequence is 4, then
Therefore, if 7 is the third number in a sequence generated by the
function
The first two numbers in the sequence are
Solving this equation, we get
The first number in the sequence is
Substituting into the first equation, we get
(Note that the two possible sequences are
Written as a product of its prime factors,
Suppose
Assume that
In this case, the divisors of
Thus, it must be that
Since
When
The refactorable numbers
Since
Thus for some integer
In this case,
This means that each of the exponents
Since
Since
For example, if
If
Thus, the smallest
Comparing these first two values of
Further, we recognize that
all remaining possible values of
where exponents are equal, smaller prime factors give smaller
values of
the greatest exponents must occur on the smallest prime factors, and
we recall that each exponent is one less than a power of 2.
In the table below, we use the above information to determine the
smallest possible values of
Further, we compare the size of each of these values of
Number of distinct prime factors of
|
Values of |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
Finally, we compare the smallest values of
Since
Let
We begin by stating a value of
For each
On the contest paper, the useful fact states that
Since
Since
Thus,
Similarly,
Therefore,
Further,
Thus for every integer