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Since
Since
Therefore,
Answer: 6
Solution 1
Since
Therefore,
Solution 2
Since
Also,
Therefore,
Answer: 40
Suppose that the two integers are
We are told that
We note that
Since 273 is a multiple of each of
The divisors of 273 are 1, 3, 7, 13, 21, 39, 91, 273.
Of these, the only pair that has a sum of 60 is 21 and 39.
Thus, the two integers are 21 and 39.
We note that since
Answer: 21 and 39
By the Pythagorean Theorem,
Let
Looking in
Thus,
Answer: 336
Since the circle has diameter
Let
Let
Join
Since quadrilateral
Thus,
Since
By the Pythagorean Theorem in
Since
Since
Consider
Since
Therefore,
By the Pythagorean Theorem in
Subtracting
Since
Answer:
We use R to represent a red marble, B to represent blue, and G to
represent green.
Since there are 15 marbles of which 3 are red, 5 are blue, and 7 are
green, there are
Since red is the first colour to have 0 remaining, the last of the 15
marbles removed must be B or G. We look at these two cases.
Suppose that the final marble is B.
The remaining 4 B’s are thus placed among the first 14 marbles. There
are
This leaves 10 open spaces. The last of these 10 marbles cannot be R,
otherwise the last G would be chosen before the last R. Thus, the last
of these 10 marbles is G.
The remaining 6 G’s are thus placed among the remaining 9 spaces. There
are
The 3 R’s are then placed in the remaining 3 open spaces.
Therefore, there are
Suppose that the final marble is G.
The remaining 6 G’s are thus placed among the first 14 marbles. There
are
This leaves 8 open spaces. The last of these 8 marbles cannot be R,
otherwise the last B would be chosen before the last R. Thus, the last
of these 8 marbles is B.
The remaining 4 B’s are thus placed among the remaining 7 spaces. There
are
The 3 R’s are then placed in the remaining 3 open spaces.
Therefore, there are
Thus, the total number of ways in which the first colour with 0
remaining is red equals
This means that the desired probability,
Answer:
Points
To find their coordinates, we set
Therefore,
We start by finding the coordinates of
Since these points lie on the horizontal line with equation
Thus,
Trapezoid
Here,
Also, the height of
Therefore, the area of
The origin,
The vertex,
When we set
The area of quadrilateral
Each of these triangles can be thought of as having a vertical base
The length of each horizontal height is 7, since the distance from each
of
Therefore, the area of
We begin by noting that
Multiplying this equation by 9 and re-arranging, we obtain the
equivalent equation
Factoring, we obtain
Since
Substituting into the original equation, we obtain
Expanding and simplifying,
Also,
Thus,
The difference between
The difference between
Since
A fact that will be important throughout this solution is that
the expresssion
In other words, when
This is true since when
Before proving a general result, we determine the number of positive
integers
When
Since each part is positive, we can square each part and obtain the
equivalent inequality
We see that
Thus, when
When
We note that
Since
When
Using similar reasoning, we obtain that this inequality is true for the
positive integers
Therefore, for
Based on these values of
From earlier, we know that the inequality
Our strategy now is to show that the inequality
Since the expression
Step (I):
We note that when
Thus, when
Step (II):
Next, we note that when
Also, when
Thus, when
Connecting (I) and (II) to (III) and (IV):
We have shown that, for all integers
Since
We have now shown the desired inequalities for
We note that if
Then
We note that every quadruple counted by
Therefore, the quadruples counted by
To determine the value of
To determine the value of
We work through pairs
Since there are thus
To answer parts (b) and (c), we determine an explicit expression
for
Suppose that
To determine an expression for
Since
This gives us two cases for the ordering of
Case 1:
Suppose that
Since
We count the number of quadruples
Suppose that
Here,
For the pair
Since
More generally, suppose that
Here,
For the pair
Since
We note that
Adding
Adding
Adding these partial sums together, we obtain the total number of
triples in this case, which is
Case 2:
Suppose that
Since
Thus, the range of values for
For each pair of values
Our strategy is to work through the possible values of
Suppose that
The base quadruple
There are
Suppose that
This gives the base quadruples
There are
Next, consider a general value of
Here,
This gives the base quadruples
For each of these values of
For a given value of
As
This sum can be re-written as
Thus, we can add these expressions for
This sum is thus
Finally, consider a general value of
Again,
In other words, when
As above, there are
As
We need to add these expressions for
We substitute
This gives
Using this formula,
when
when
when
which agrees with the specific results that we found earlier.
Thus,
(Note that we could have found this value by calculating the value of
Lastly, we find two even positive integers
To do this, we note first that
Therefore, we need to find two positive integers
We can assume that
When
Each of these expressions is even, since each has a factor of 4 in
it.
Therefore, we need to determine 2 small enough values of
Since 337 is a larger prime, we work by looking for factors of 337 first
and then checking for factors of 3.
The first two positive multiples of 337 are 337 and 674.
We can check by working through the factors of these three expressions
that it is impossible for one of the factors to equal 337 and for
another factor in the same expression to be a multiple of 3.
When
Therefore, when
When
Therefore, when
This gives us the answer to (c): two even integers