Wednesday, November 23, 2016
(in North America and South America)
Thursday, November 24, 2016
(outside of North American and South America)
©2016 University of Waterloo
Solution 1
We re-write each of the fractions using the common denominator 24:
Solution 2
We note that
The smallest of these four fractions is the one which is equal to 1 minus the largest of the unit fractions
So we need to compare
Since
Answer:
Solution 1
From the graph, the numbers of students whose birthday in 2016 is on Sunday, Monday, Wednesday, Thursday, Friday, and Saturday are 4, 4, 2, 2, 8, and 6, respectively.
Therefore, the number of students whose birthday is not on a day beginning with “T” is
Since 25% of the students have their birthday in 2016 on a day beginning with “T”, then
Since 75% is equivalent to
Since 2 students have their birthday on Thursday and 8 students (25% of the class) have their birthday in 2016 on a day beginning with “T”, then the number of students with a birthday on Tuesday is
Solution 2
Suppose that there were
Then there are
Also, there are
We note that
Since 25% of the students in Ms. Gupta’s class have their birthday in 2016 on a day beginning with “T", then the total number of students is 4 times the number of students with a birthday on a day beginning with the letter “T".
Thus,
This gives
In other words, there were 6 students with their birthday in 2016 on a Tuesday.
Answer: 6
Since there are 12 hurdles, then there are 11 gaps between the hurdles, each of length
The gap before the first hurdle is 50 metres and the gap after the last hurdle is 55 metres.
Since the race is 600 metres long, then
Answer:
Solution 1
Dina’s machine works by taking an input, multiplying by 2 and then subtracting 3.
If we have the output and want to obtain the input, we must reverse these operations and so we take the output, add 3 and then divide by 2.
Starting with the second output
Since the second input was the first output, then the first output was
Starting with the first output
Therefore, the first input was
Solution 2
Starting with the first input
Dina inputs this output back into the machine.
The machine multiplies the input by 2 to obtain
Since the second output is
Answer:
Solution 1
Since the distance from
Since the circumference of the circle is
In other words,
Similarly, since the distance from
We now make two charts that list possible distances travelled by
Distance for |
Time (s) (rounded to the nearest hundredth) |
---|---|
8 | 2.67 |
48 | 16 |
88 | 29.33 |
128 | 42.67 |
168 | 56 |
208 | 69.33 |
248 | 82.67 |
288 | 96 |
Distance for |
Time (s) (rounded to the nearest hundredth) |
---|---|
16 | 4.57 |
56 | 16 |
96 | 27.43 |
136 | 38.86 |
176 | 50.29 |
216 | 61.71 |
256 | 73.14 |
296 | 84.57 |
336 | 96 |
In each case, we determine the time by taking the distance travelled and dividing by the appropriate constant speed (3 m/s for
From the tables, we see that
Therefore,
Solution 2
The circumference of the circle is
Since
Similarly, since
Suppose that
Since
Since
For
Multiplying the first of these equations by 7 and the second of these equations by 6, we obtain
Equating the expressions equal to
Manipulating this equation, we obtain the equivalent equations
Since we are looking for the second smallest possible value of
We do this by listing values of
Multiple of 7? | ||
---|---|---|
1 | 7 | Y |
2 | 13 | N |
3 | 19 | N |
4 | 25 | N |
5 | 31 | N |
6 | 37 | N |
7 | 43 | N |
8 | 49 | Y |
Answer: 96
Every line is determined by any two points on the line.
We use
We use
We determine the slope and
The first point that we choose is the point,
We note that the line of reflection can be re-written as
Since
Since
Since
In other words,
To find the
Since
Thus, the coordinates of
Consider next the point
Let
Since
Since
Therefore,
Since
Thus,
In addition to
The midpoint of
Since this point lies on the line with equation
Thus,
Therefore,
The slope of
Therefore,
Since
Therefore, the slope of
Answer: Slope is
Also,
Since
We extend
Then figure
In fact,
Also,
Therefore, the area of
We draw lines through
The area of
Here,
Also,
Thus, its area is
Thus, its area is
Finally, this tells us that the area of
(This can also be solved using a result called Pick’s Theorem, which involves counting the number of grid dots that occur on the boundary and in the interior of the shape.)
One way of arranging these integers is
The positive differences between the pairs of adjacent integers are
There are many other ways of arranging these integers so that the desired property is true.
(We note that this property is equivalent to arranging the five integers so that no two consecutive integers are adjacent.)
In any arrangement, the integer 10 must be placed next to at least one other integer.
From the list
The positive differences between 10 and these integers are 9 and 10.
In other words, the positive difference between 10 and every other integer in the list is less than or equal to 10.
Therefore, in any arrangement, there must be a positive difference that is at most 10, and so
Consider the following arrangement:
Consider the integer 14, which is the middle integer in the list
The maximum possible positive difference between 14 and another number from this list is 13.
Therefore, in any arrangement, there must be a positive difference that is no larger than 13.
This tells us that
To show that the maximum possible value of
Here is such an arrangement:
To see why the Murray number of 6 is 12, we need to argue that
there exist one or more distinct integers greater than 6 and less than or equal to
First, we note that
Second, for 6 times a product of one or more integers to be a perfect square, this product must include an odd number of factors of 3. This is because 6 includes exactly an odd number of factors of 3 and a perfect square must include an even number of factors of 3. For a product to include an odd number of factors of 3, not all of the integers in the product can include an even number of factors of 3, so one of the integers in the product must include an odd number of factors of 3. Since the first two multiples of 3 larger than 6 are 9 and 12, and 9 includes an even number of factors of 3, then the product must include a multiple of 3 that is at least as large as 12. In other words, we need to have
These two statements explain why 12 is the Murray number of 6.
The Murray number of 8 is 15.
While no justification is required for full marks on this part, we present a justification modelled after the solution in (a).
First,
The statement in the previous sentence shows that
Is it possible that
Since 8 includes an odd number of factors of 2, then the Murray product must include another integer with an odd number of factors of 2.
Since
If 10 is in the Murray product, then, because 10 includes an odd number of factors of 5, we need to include another multiple of 5, which must be at least 15.
If 14 is in the Murray product, then, because 14 includes an odd number of factors of 7, we would need to include another multiple of 7, which must be at least 21, which is too large.
Therefore, it is not possible that
Let
We see that
The product
This shows that the Murray number of
Since there are infinitely many possible values for
(There are many other ways to solve this problem.)
Solution 1
Let
First, we prove that the Murray number of
Since
Thus, the Murray number
Next, we prove that the Murray number of
To do this, we prove that the Murray number of
Since the Murray number of
Case 1: Can the Murray number of
If this were the case, then there would exist one or more distinct integers greater than
Since
We note that
Since
Also,
In other words,
Since
This tells us that
Case 2: Can the Murray number of
If this were the case, then there would exist one or more distinct integers greater than
Since
Now,
Thus,
To conclude our argument, we must show that
To show this, we will use the following three facts:
(F1) The difference between two positive perfect squares cannot equal 1.
Suppose
and is the smaller of a pair of positive perfect squares.
The closest larger perfect square is, and so the smallest possible difference between and a larger perfect square equals or or which is at least 3 since .
Therefore, the difference between two positive perfect squares cannot equal 1.
(F2) The largest possible common divisor that two of
Suppose that two integers are multiples of
.
Then their difference must also be a multiple of.
Among the three integers, the possible differences between pairs are 1 and 2.
Thus, the possible positive common divisors are the positive divisors of 1 and 2, which are 1 and 2.
Therefore, the largest possible common divisor that two ofcan share is 2.
(F3) If the product of two or more positive integers is a perfect square and each pair of these positive integers has no common divisor larger than 1, then each of these positive integers must be a perfect square.
Consider a prime number
that is a factor of the product.
Since the product is a perfect square, then the product includes an even number of factors of.
Nowcannot divide into more than one of the integers that make up the product, so it must be included an even number of times in one of the integers and cannot be included in any of the other integers.
Since this is true for any prime numberthat is a factor of the product, then each of integers that make up the product must be a perfect square.
We now consider
If
Since the maximum possible common divisor of any pair of these is 2 (by (F2)) and only one of these integers is even, then no pair of these integers has a common divisor larger than 1.
By (F3), each of
But by (F1), no two perfect squares differ by 1.
Thus,
If
By the arguments above, every prime factor other than 2 of one of these three integers is not included in either of the other integers. Also, this factor must be included an even number of times in the integer in which it appears.
Since
By (F1), neither
Thus,
In this case, we have
Thus,
Therefore, the Murray number of
Solution 2
Let
First, we prove that the Murray number of
Since
Thus, the Murray number
To show that the Murray number of
We note that
and are consecutive perfect squares.
Sinceis a positive integer then, .
Also,and
In other words,.
Sinceand are between two consecutive perfect squares, then neither nor can be a perfect square.
To prove that the Murray number of
Since the Murray number of
Case 1: Can the Murray number of
If this were the case, then there would exist one or more distinct integers greater than
Since
Applying the fact above with
Case 2: Can the Murray number of
If this were the case, then there would exist one or more distinct integers greater than
Since
Applying the fact above with
To conclude our argument, we must show that
Suppose that
If
Thus,
This means that there is a prime number
Since
If
Thus,
This means that
Thus,
In this case,
Since
Finally, this means that
Therefore, the Murray number of