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2016 Canadian Intermediate
Mathematics Contest
Solutions

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


PART A

  1. Solution 1
    We re-write each of the fractions using the common denominator 24: 23=162434=182456=202458=15241112=2224 The smallest of these is 1524=58.

    Solution 2
    We note that 23=113 and 34=114 and 56=116 and 1112=1112.
    The smallest of these four fractions is the one which is equal to 1 minus the largest of the unit fractions 13,14,16,112. This is 23=113.
    So we need to compare 23 and 58, the smaller of which will be the smallest of the five fractions in the list.
    Since 23=0.60.667 and 58=0.625, then 58 is the smallest of the five fractions.

    Answer: 58

  2. 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 4+4+2+8+6=24.
    Since 25% of the students have their birthday in 2016 on a day beginning with “T”, then 100%25%=75% of the students have their birthday on a day not beginning with “T”.
    Since 75% is equivalent to 34, then 24 students represent 34 of the class, and so 8 students represent 14 or 25% of the class.
    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 82=6.

    Solution 2
    Suppose that there were t students with their birthday on Tuesday.
    Then there are t+2 students born on a day beginning with “T".
    Also, there are 4+4+t+2+2+8+6=26+t students in Ms. Gupta’s class.
    We note that 25% is equivalent to 14.
    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, 26+t=4(t+2).
    This gives 26+t=4t+8 or 18=3t and so t=6.
    In other words, there were 6 students with their birthday in 2016 on a Tuesday.

    Answer: 6

  3. Since there are 12 hurdles, then there are 11 gaps between the hurdles, each of length d metres.
    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 50+11d+55=600 or 11d=495 and so d=45.

    Answer: d=45

  4. 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 35, we obtain 35+3=32 and then 322=16, which was the second input.
    Since the second input was the first output, then the first output was 16.
    Starting with the first output 16, we obtain 16+3=13 and then 132, which was the first input.
    Therefore, the first input was 132 or 6.5.

    Solution 2
    Starting with the first input x, Dina’s machine multiplies it by 2 to obtain 2x and then subtracts 3 to obtain 2x3.
    Dina inputs this output back into the machine.
    The machine multiplies the input by 2 to obtain 2(2x3)=4x6 and then subtracts 3 to obtain 4x9.
    Since the second output is 35, then 4x9=35 or 4x=26 and so x=264=132 or 6.5.

    Answer: 132

  5. Solution 1
    Since the distance from P’s initial position to X along the circle is 8 m, then after travelling 8 m, P will be at X.
    Since the circumference of the circle is 8+16+16=40 m, then after travelling any additional multiple of 40 m, P will again be at X.
    In other words, P is at X after travelling 8 m, 48 m, 88 m, 128 m, 168 m, and so on.
    Similarly, since the distance from Q’s initial position to X along the circle is 16 m, then Q is at X after travelling 16 m, 56 m, 96 m, 136 m, 176 m, and so on.
    We now make two charts that list possible distances travelled by P and Q to arrive at X and the corresponding times that these distances take. We are looking for the same lengths of time to appear in both tables, as this would indicate times when P and Q are both at X.

    Distance for P (m) 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 Q (m) 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 P and 3.5 m/s for Q). For example, Q travels 56 m in 56 m3.5 m/s=16 s.
    From the tables, we see that P and Q meet at X after 16 s (having travelled 163=48 m and 163.5=56 m, respectively) and then again after 96 s (having travelled 963=288 m and 963.5=336 m, respectively).
    Therefore, P and Q meet at X for the second time after 96 s.

    Solution 2
    The circumference of the circle is 8+16+16=40 m.
    Since P starts 8 m from X, then P is at X after moving 8+40p metres, for each integer p0. (The variable p represents the number of complete times that P has moved around the circle after reaching X for the first time.)
    Similarly, since Q starts 16 m from X, then Q is at X after moving 16+40q metres, for each integer q0.
    Suppose that P and Q meet at X after moving for t seconds. We want to determine the second smallest possible value of t.
    Since P moves at 3 m/s, then after t seconds, P has moved 3t metres.
    Since Q moves at 3.5 m/s, then after t seconds, Q has moved 3.5t metres.
    For P and Q to be at X after t seconds, we need 3t=8+40p and 3.5t=16+40q for some integers p,q0.
    Multiplying the first of these equations by 7 and the second of these equations by 6, we obtain 21t=56+280p and 21t=96+240q.
    Equating the expressions equal to 21t, we obtain 56+280p=96+240q.
    Manipulating this equation, we obtain the equivalent equations 280p=40+240q and 7p=1+6q.
    Since we are looking for the second smallest possible value of t, then we look for the second smallest pair of integers p and q with p,q0 that satisfy this equation.
    We do this by listing values of q and the corresponding values of 1+6q until we obtain the second q for which 1+6q is a multiple of 7:

    q 1+6q 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
    Therefore, the second time at which P and Q meet at X is when p=7 and q=8.
    When p=7, we obtain 3t=8+40p=8+40(7)=288 and so t=96.

    Answer: 96

  6. Every line is determined by any two points on the line.
    We use L to denote the line with equation x+y=1.
    We use to denote the line with equation y=kx and to denote the line obtained after reflecting in L.
    We determine the slope and y-intercept of by finding two points on and then using these points to determine the slope, equation, and y-intercept.
    The first point that we choose is the point, P, of intersection between the original line, , y=kx, and the line of reflection, L.
    We note that the line of reflection can be re-written as y=x+1, which has slope 1.
    Since k1, then is not parallel to L and so there is indeed a point of intersection.
    Since P is on , then its reflection will be on the resulting line.
    Since P is on L, then its reflection will be itself.
    In other words, P is also on .
    To find the x-coordinate of P, we substitute y=kx into x+y=1 to obtain x+kx=1 or x(k+1)=1 and so x=1k+1.
    Since P is on , which has equation y=kx, then y=k1k+1=kk+1.
    Thus, the coordinates of P are (1k+1,kk+1).
    Consider next the point O(0,0) which is on .
    Let Q be the reflection of O in L.
    Since Q is the reflection of O in L, then OQ is perpendicular to L.
    Since L has slope 1, then any line perpendicular to L has slope equal to 11=1.
    Therefore, OQ has slope 1.
    Since O is the origin (0,0), then the line through O and Q has equation y=x.
    Thus, Q has coordinates (q,q) for some real number q.
       

    A graph with the origin (point O), a dotted line between point O and point Q which is situated 45 degrees from the positive x and y-axis. There is also a perpendicular line intersecting the middle of the dotted line as well as the positive x and y-axis. A right angle is shown between these two lines to show orthogonality.
    In addition to OQ being perpendicular to L, we also need the midpoint of OQ to lie on L.
    The midpoint of O(0,0) and Q(q,q) has coordinates (12(0+q),12(0+q))=(12q,12q).
    Since this point lies on the line with equation x+y=1, then 12q+12q=1 or q=1.
    Thus, Q has coordinates (1,1).
    Therefore, passes through Q(1,1) and P(1k+1,kk+1).
    The slope of is thus 1kk+111k+1=k+1k+1kk+1k+1k+11k+1=1k+1kk+1=1k. (Note that k0.)
    Therefore, has equation y=1kx+b for some b.
    Since Q(1,1) lies on , then 1=1k+b or b=k1k.
    Therefore, the slope of is 1k and its y-intercept is k1k.

    Answer: Slope is 1k; y-intercept is k1k

PART B

  1. We now consider n(n+1)(n+2) and assume that it is a perfect square.
    If n is odd, then n+1 is even and n+2 is odd.
    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 n,n+1,n+2 must be a perfect square.
    But by (F1), no two perfect squares differ by 1.
    Thus, n cannot be odd.
    If n is even, then n+1 is odd and n+2 is even.
    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 n+1 is odd, it has no factors of 2, and so must be a perfect square since all of its other prime factors will occur in pairs.
    By (F1), neither n nor n+2 can be a perfect square.
    Thus, n=2b2 and n+2=2c2 for some positive integers b and c. This is because each is even, any other prime factor occurs in pairs, and neither can include an even number of factors of 2 (otherwise it would be a perfect square).
    In this case, we have 2=(n+2)n=2c22b2 and so c2b2=1, which contradicts (F1).
    Thus, n cannot be even, and so n(n+1)(n+2) cannot be a perfect square.
    Therefore, the Murray number of n exists and cannot be n+1 or n+2, and so must be greater than or equal to n+3.

    Solution 2
    Let n be a positive integer.
    First, we prove that the Murray number of n exists.
    Since n×4n=4n2=(2n)2, then there does exist a set of one or more distinct integers larger than n whose product times n is a perfect square.
    Thus, the Murray number M of n exists and satisfies n<M4n. The fact that n×4n is a perfect square does not necessarily tell us that 4n is the Murray number. Since the integer 4n has the correct property to be the Murray number, then the question that must be answered is whether or not there is an integer smaller than 4n (but still larger than n) with this property; if not, then 4n is the Murray number.
    To show that the Murray number of n is at least n+3, we first show that if j is a positive integer, then neither j(j+1) nor j(j+2) can be a perfect square:

    We note that j2 and (j+1)2=j2+2j+1 are consecutive perfect squares.
    Since j is a positive integer then, j2=j×j<j(j+1)<j(j+2).
    Also, j(j+1)<(j+1)(j+1)=(j+1)2 and j(j+2)=j2+2j<j2+2j+1=(j+1)2
    In other words, j2<j(j+1)<j(j+2)<(j+1)2.
    Since j(j+1) and j(j+2) are between two consecutive perfect squares, then neither j(j+1) nor j(j+2) can be a perfect square.

    To prove that the Murray number of n is greater than or equal to n+3, we prove that the Murray number of n cannot be less than n+3.
    Since the Murray number of n is greater than n, then the only possibilities less than n+3 are n+1 and n+2.
    Case 1: Can the Murray number of n be n+1?
    If this were the case, then there would exist one or more distinct integers greater than n and less than or equal to n+1 whose product times n is a perfect square.
    Since n+1 is the only integer that satisfies these restrictions, then for n+1 to be the Murray number of n, it must be the case that n(n+1) is a perfect square.
    Applying the fact above with j=n tells us that this cannot be true, and so n+1 cannot be the Murray number of n.
    Case 2: Can the Murray number of n be n+2?
    If this were the case, then there would exist one or more distinct integers greater than n and less than or equal to n+2 whose product times n is a perfect square.
    Since n+1 and n+2 are the only integers that satisfy these restrictions, then for n+2 to be the Murray number of n, it must be the case that n(n+2) is a perfect square or n(n+1)(n+2) is a perfect square. (The product must include itself n+2 otherwise the Murray number would be less than n+2. Also, the product either includes n+1 or it does not.)
    Applying the fact above with j=n shows us that n(n+2) cannot be a perfect square.
    To conclude our argument, we must show that n(n+1)(n+2) cannot be a perfect square.
    Suppose that n(n+1)(n+2) is a perfect square. (We will show that this cannot be true.)
    If n is itself a perfect square, then n(n+1)(n+2)n=(n+1)(n+2) is also a perfect square. But this cannot be true, which we see by applying the fact above with j=n+1.
    Thus, n cannot itself be a perfect square in this case.
    This means that there is a prime number p which divides n an odd number of times. (This is true because an integer larger than 1 is a perfect square exactly when every prime factor is included an even number of times.)
    Since n(n+1)(n+2) is a perfect square, then p divides into n(n+1)(n+2) an even number of times, and so p divides into n+1 or into n+2.
    If n+1 and n are both multiples of p, then their difference is a multiple of p. In other words, 1 is a multiple of p, which is not possible.
    Thus, n+2 is a multiple of p. But again, p must divide into the difference between n+2 and n, which is 2. In other words, we must have p=2.
    This means that p=2 is the only prime number that divides into each of n and n+2 an odd number of times, and so each of n and n+2 equals 2 times a perfect square.
    Thus, n=2b2 and n+2=2c2 for some positive integers b and c with b<c.
    In this case, 2=(n+2)n=2c22b2, and so 1=c2b2=(c+b)(cb).
    Since c and b are positive integers with c>b and (c+b)(cb)=1, then c+b=cb=1, which gives c=1 and b=0, which is a contradiction.
    Finally, this means that n(n+1)(n+2) cannot be a perfect square, and so n+2 is not the Murray number of n.
    Therefore, the Murray number of n exists and cannot be n+1 or n+2, and so must be greater than or equal to n+3.