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(outside of North American and South America)
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Since
Since
Factoring, we obtain
Solution 1
Manipulating algebraically, we obtain the following equivalent equations:
Solution 2
Manipulating algebraically, we obtain the following equivalent equations:
Using a calculator, we see that
To determine this integer without using a calculator, we can let
Then
Before the price increase, the total cost of 2 small cookies and 1 large cookie is
10% of $1.50 is
5% of $2.00 is
After the price increase, the total cost of 2 small cookies and 1 large cookie is
The percentage increase in the total cost is
Suppose that Rayna’s age is
Since Qing is twice as old as Rayna, Qing’s age is
Since Qing is 4 years younger than Paolo, Paolo’s age is
Since the average of their ages is 13 years, we obtain
Therefore, Rayna is 7 years old, Qing is 14 years old, and Paolo is 18 years old.
(Checking, the average of 7, 14 and 18 is
The length of
In a similar way, we can see that
Therefore, the perimeter of
(We can also see that if
Solution 1
Suppose that
Since
Since
The slope of the line segment joining
Solution 2
Since
Therefore, the slope of
Since
Thus, the equation of the parabola is
The
The
This means that
Therefore,
Finally, the area of
We obtain successively
Since the list
If
If
Thus, the possible values for
Solution 1
Suppose that
Step 1 moves
Step 2 moves
Step 3 moves
Thus,
Step 1 moves
Step 2 moves
Step 3 moves
Thus,
Continuing this process,
Since the coordinates of
Solution 2
We work backwards from
To do this, we undo the Steps of the process
Since Step 3 reflects a point in the
Since Step 2 translates a point 2 units upwards, its inverse translates a point 2 units downwards.
Since Step 1 reflects a point in the
Applying these inverse steps to
Thus,
Applying the inverse steps to
Thus,
If we apply these steps two more times, we will see that
Two more applications tell us that
Therefore, the coordinates of
We begin by determining the length of
Since
Since
Join
Since
Consider
Using ratios of side lengths,
Next, we determine
Now,
Suppose that
Since
Join
Since
Also,
Since
Since
Now
This means that
Finally,
This means that
Solution 1
Since the sequence
Since the sum of all
Solution 2
Suppose that the arithmetic sequence with
Then
Since the sum of the
Substituting, we get
Since the sum of a geometric sequence with first term
If
An alternate way of arriving at the equation
Dividing the second equation by the first, we obtain
Victor stops when there are either 2 green balls on the table or 2 red balls on the table.
If the first 2 balls that Victor removes are the same colour, Victor will stop.
If the first 2 balls that Victor removes are different colours, Victor does not yet stop, but when he removes a third ball, its colour must match the colour of one of the first 2 balls and so Victor does stop.
Therefore, the probability that he stops with at least 1 red ball and 1 green ball on the table is equal to the probability that the first 2 balls that he removes are different colours.
Also, the probability that the first 2 balls that he removes are different colours is equal to 1 minus the probability that the first 2 balls that he removes are the same colour.
The probability that the first two balls that Victor draws are both green is
The probability that the first two balls that Victor draws are both red is
Thus, the probability that the first two balls that Victor removes are the same colour is
Using the definition of
Thus,
Using the definition of
Therefore,
Since
Suppose that the integers in the first row are, in order,
Using these, we calculate the integer in each of the boxes below the top row in terms of these variables, using the rule that each integer is the product of the integers in the two boxes above:
Therefore,
Next, we determine the prime factorization of the integer
Since the right side is not divisible by 7, none of
Thus,
The only one of these integers divisible by 5 is 5 itself.
Since
Case 1:
Here,
Since
Case 1a:
Here,
This gives
Trying the four possible values of
Case 1b:
Case 1c:
Here,
This gives
Trying the four possible values of
Case 1d:
Case 2:
Therefore, there are 8 ways in which the integers can be chosen and placed in the top row to obtain the desired integer in the bottom box.
Let
For each integer
Therefore,
(In particular,
Thus,
Since there are 200 sets of parentheses, we obtain
Since
When
Therefore, we want to find all pairs of integers
If
If
If
If
Similarly, if
Therefore, the solutions are
Suppose that
Then
How could we come up with this? One way to do this would be trying some small values of
1 | 1 | 4 | 2 | 0 |
2 | 1 | 7 | 3 | 1 |
3 | 1 | 12 | 4 | 2 |
1 | 2 | 13 | 3 | -1 |
2 | 2 | 16 | 4 | 0 |
3 | 2 | 21 | 5 | 1 |
Suppose that
If
If
If
Therefore, in all cases, for all integers
As in (b), trying some small cases might help us make a guess of possible expressions for
1 | 1 | 1 | 1 | 0 |
2 | 1 | 3 | 0 | 1 |
3 | 1 | 7 | 2 | 1 |
4 | 1 | 13 | 1 | 2 |
1 | 2 | 3 | 0 | 1 |
2 | 2 | 4 | 1 | 1 |
3 | 2 | 7 | 2 | 1 |
4 | 2 | 12 | 3 | 1 |
5 | 3 | 19 | 4 | 1 |
2 | 1 | 3 | 0 | 1 |
4 | 1 | 13 | 1 | 2 |
2 | 2 | 4 | 1 | 1 |
4 | 2 | 12 | 3 | 1 |
1 | 2 | 3 | 0 | 1 |
3 | 2 | 7 | 2 | 1 |
4 | 2 | 12 | 3 | 1 |
1 | 1 | 1 | 1 | 0 |
3 | 1 | 7 | 2 | 1 |
5 | 3 | 19 | 4 | 1 |
We label the centres of the outer circles, starting with the circle labelled
Join
When two circles are tangent, the distance between their centres equals the sum of their radii.
Thus,
Let
By congruent triangles,
Since
Consider
By the cosine law in
Let the centre of the middle circle be
Join
Using a similar argument as in (a), we see that
This means that
Since
By the Pythagorean Theorem,
Manipulating algebraically, the following equations are equivalent:
Also, if real numbers
In other words, the diagram can be drawn exactly when
Suppose that
Determining the value of
Since
Since
Also, since
Therefore, every pair
Similarly, every divisor pair
In other words,
Therefore, we want to determine all positive integers
Suppose that the prime factorization of
Then
The first factor in this product is even and each factor after the first is odd.
Therefore, this product is a multiple of 4 exactly when
This is true exactly when
In other words, the number of positive divisors of
Finally, this means that the positive integers for which