Tuesday, February 27, 2018
(in North America and South America)
Wednesday, February 28, 2018
(outside of North American and South America)
©2017 University of Waterloo
Since
Answer:(E)
The
For half of these to be shaded, 10 must be shaded.
Since 3 are already shaded, then
Answer: (C)
Since the number line between 0 and 2 is divided into 8 equal parts, then the portions between 0 and 1 and between 1 and 2 are each divided into 4 equal parts.
In other words, the divisions on the number line mark off quarters, and so
Answer: (D)
Since
Alternatively, we can note that
Answer: (D)
The entire central angle in a circle measures
In the diagram, the central angle which measures
Therefore, the area of the sector is
Answer: (B)
For any value of
When
Answer: (B)
We want to calculate the percentage increase from 24 to 48.
The percentage increase from 24 to 48 is equal to
Alternatively, since 48 is twice 24, then 48 represents an increase of 100% over 24.
Answer: (D)
A line segment joining two points is parallel to the
Here, this means that
(We can check that when
Answer: (B)
Since
Multiplying by 3, we obtain
The average of
Answer:(E)
Of the given uniform numbers,
11 and 13 are prime numbers
16 is a perfect square
12, 14 and 16 are even
Since Karl’s and Liu’s numbers were prime numbers, then their numbers were 11 and 13 in some order.
Since Glenda’s number was a perfect square, then her number was 16.
Since Helga’s and Julia’s numbers were even, then their numbers were 12 and 14 in some order. (The number 16 is already taken.)
Thus, Ioana’s number is the remaining number, which is 15.
Answer: (D)
Solution 1
The large square has side length 4, and so has area
The small square has side length 1, and so has area
The combined area of the four identical trapezoids is the difference in these areas, or
Since the 4 trapezoids are identical, then they have equal areas, each equal to
Solution 2
Suppose that the height of each of the four trapezoids is
Since the side length of the outer square is 4, then
Each of the four trapezoids has parallel bases of lengths 1 and 4, and height
Therefore, the area of each is
Answer: (D)
We are told that 1 Zed is equal in value to 16 Exes.
We are also told that 2 Exes are equal in value to 29 Wyes.
Since 16 Exes is 8 groups of 2 Exes, then 16 Exes are equal in value to
Thus, 1 Zed is equal in value to 232 Wyes.
Answer: (C)
The problem is equivalent to determining all values of
The divisors of 3 are
If
Answer: (A)
Solution 1
The line segment with endpoints
This means that starting at
These points are
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between
Consider the point on this line segment with
Since this point has
In a similar way, the points on the line segment with
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
Solution 2
The line segment with endpoints
Since the line passes through
Suppose that a point
Since
This means that
Since
This leads to the points listed in Solution 1, and justifies why there are no additional points.
Therefore, there are 6 such points.
Answer: (E)
Since
Since
Since
Since
This means that
Answer: (A)
Elisabeth climbs a total of 5 rungs by climbing either 1 or 2 rungs at a time.
Since there are only 5 rungs, then she cannot climb 2 rungs at a time more than 2 times.
Therefore, she must climb 2 rungs either 0, 1 or 2 times.
If she climbs 2 rungs 0 times, then each step consists of 1 rung and so she climbs
If she climbs 2 rungs 1 time, then she climbs
But she can climb these numbers of rungs in several different orders.
Since she takes four steps, she can climb 2 rungs as any of her 1st, 2nd, 3rd, or 4th step.
Putting this another way, she can climb
There are
If she climbs 2 rungs 2 times, then she climbs
Again, she can climb these numbers of rungs in several different orders.
Since she takes three steps, she can climb 1 rung as any of her 1st, 2nd or 3rd step.
Putting this another way, she can climb
There are
In total, there are
Answer: (E)
Since
This means that
Therefore,
(A specific example of
This gives
We should also note that if
Since
Also, the denominator of our desired expression
Therefore, if
Answer: (B)
Solution 1
The lines with equations
The lines with equations
Since the quadrilateral has two sets of parallel sides, it is a parallelogram.
Thus, its area equals the length of its base times its height.
We consider the vertical side along the
Since the two sides of slope 1 have
Since the parallel vertical sides lie along the lines with equations
Therefore, the area of the quadrilateral is
Solution 2
The lines with equations
The lines with equations
The lines with equations
The lines with equations
We draw horizontal lines through
The area of the quadrilateral equals the area of the large rectangle with vertices
This rectangle has side lengths
The two triangles are right-angled and have bases of length
Therefore, the area of the quadrilateral is
Answer: (E)
Suppose that
Since the area of the overlapped region is
Since the area of the overlapped region is
Therefore,
Multiplying both sides by 25 to clear denominators, we obtain
Dividing both sides by 3, we obtain
This means that the ratio of the area of the small circle to the area of the large circle is
Answer: (D)
When the product of the three integers is calculated, either the product is a power of 2 or it is not a power of 2.
If
Therefore, we can calculate
For the product of the three integers to be a power of 2, it can have no prime factors other than 2. In particular, this means that each of the three integers must be a power of 2.
In each of the three sets, there are 3 powers of 2 (namely, 2, 4 and 8) and 2 integers that are not a power of 2 (namely, 6 and 10).
This means that the probability of choosing a power of 2 at random from each of the sets is
Since Abigail, Bill and Charlie choose their numbers independently, then the probability that each chooses a power of 2 is
In other words,
Answer: (C)
Since each of
This can only happen if
But
So if
But
Therefore, we cannot have
The next largest possible value of
We can construct the diagram with this value of
Therefore, the maximum possible value of
Answer: (B)
For the given expression to be equal to an integer, each prime factor of the denominator must divide out of the product in the numerator.
In other words, each prime number that is a factor of the denominator must occur as a factor at least as many times in the numerator as in the denominator.
We note that
Also,
Therefore, the given expression is equal to
We count the number of times that each of 5, 3 and 2 is a factor of the numerator.
The product equal to
Each of these factors includes 1 factor of 5, except for 25 which includes 2 factors.
Therefore, the numerator includes 7 factors of 5.
For the numerator to include at least as many factors of 5 as the denominator, we must have
Since
The product equal to
Seven of these include exactly one factor of 3, namely
Two of these include exactly two factors of 3, namely
One of these includes exactly three factors of 3, namely 27.
Therefore, the numerator includes
For the numerator to include at least as many factors of 3 as the denominator, we must have
Since
If
Since
We note that if
Answer: (A)
To determine the volume of the prism, we calculate the area of its base and the height of the prism.
First, we calculate the area of its base.
Any cross-section of the prism parallel to its base has the same shape, so we take a cross-section 1 unit above the base.
Since each of the spheres has radius 1, this triangular cross-section will pass through the centre of each of the spheres and the points of tangency between the spheres and the rectangular faces.
Let the vertices of the triangular cross-section be
Join
We determine the length of
Note that
Also,
Since
Therefore,
Since
Thus,
Since
Now
This means that
Since
Similarly,
Therefore,
This means that
To calculate the area of
Since
Since
Therefore, the area of
This means that the area of the base of the prism (which is
Now we calculate the height of the prism.
Let the centre of the top sphere be
The vertical distance from
Similarly, the vertical distance from the bottom face to the plane through
To finish calculating the height of the prism, we need to determine the vertical distance between the cross-section through
The total height of the prism equals
Since the four spheres touch, then the distance between any pair of centres is the sum of the radii, which is 2.
Therefore,
We need to calculate the height of this tetrahedron.
Join
By symmetry,
Let
Join
Since
Since
Finally,
Thus,
This means that the height of the prism is
The volume of the prism is equal to the area of its base times its height, which is equal to
Of the given answers, this is closest to 47.00.
Answer: (E)
Since there must be at least 2 gold socks (
This means that there are 2 gold socks left to place. There are
These 2 socks can either be placed together in one location or separately in two locations.
If the 2 socks are placed together, they can be placed in any one of the
The other possibility is that the 2 gold socks are placed separately in two of these
There are
Since these two socks are identical, we have double-counted the total number of possibilities, and so there are
In total, there are
We want to determine the smallest positive integer
This is equivalent to determining the smallest positive integer
We note that
When
When
Since
The sum of the digits of
Answer: (A)
Since the terms in each such sequence can be grouped to get both positive and negative sums, then there must be terms that are positive and there must be terms that are negative.
Since the 15 terms have at most two different values and there are terms that are positive and terms that are negative, then the terms have exactly two different values, one positive and one negative.
We will call these values
Consider one of these sequences and label the terms
Since the sum of eleven consecutive terms is always negative, then
Since
The six term condition also tells us that
Since
The six term condition also tells us that
Since
We can repeat this argument by shifting all of the terms one further along in the sequence.
Starting with
By continuing to shift one more term further along twice more, we obtain
So far this gives the sequence
Starting with
Shifting back to the left and repeating this argument gives
This means that the sequence has the form
In fact, it must be the case that both of these terms equal
To see this, suppose that
In this case, the sum of the first 6 terms in the sequence is
Also, the sum of the first 11 terms in the sequence is
But
We would obtain the same result if we considered the possibility that
Therefore,
We are told that
Also, the sum of each group of 11 consecutive terms is
We are told that
We have now changed the original problem into an equivalent problem: count the number of pairs
Suppose that
From
We can now make a chart that enumerates the values of
Possible |
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Answer: (E)