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2022 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 16, 2022
(in North America and South America)

Thursday, November 17, 2022
(outside of North American and South America)

©2022 University of Waterloo


PART A

  1. Since the area of one square face of the cube is 16 cm2, the edge length of the cube is 16 cm2 which equals 4 cm.
    Since the edge length of the cube is 4 cm, its volume is (4 cm)3=43 cm3=64 cm3.
    Thus, V=64.

    Answer: 64

  2. Since ADC is isosceles with AD=DC, we have DAC=DCA=40°.
    Since the sum of the measures of the angles in a triangle is 180°, we have ADC=180°DACDCA=180°40°40°=100° Since a straight angle measures 180°, we have ADB=180°ADC=180°100°=80°.
    Since ADB is isosceles with AD=DB, we have DAB=DBA and so DAB=12(180°ADB)=12(180°80°)=50° Finally, BAC=DAB+DAC=50°+40°=90°.

    We note that students who know about “exterior angles” could shorten this solution by noting that, for example, ADB=DAC+DCA=40°+40°=80°.
    Additionally, some students may note that since DA=DB=DC, then D is the centre of the circle that passes through A, B and C (that is, the circumcircle of ABC). Since BC will be a diameter of this circle with A a point on its circumference, then BAC=90°.

    Answer: 90°

  3. Since Marie-Pascale solves 4 math problems per day and solves 72 problems in total, this takes her 72÷4=18 days.
    Since Kaeli solves 54 more problems than Marie-Pascale, she solves 72+54=126 problems.
    Since Kaeli solves x problems per day over 18 days and solves 126 problems in total, then x=12618=7.

    Answer: 7

  4. Solution 1

    Since cd=715, then c=715d.
    Since cb=15, then b=5c=5×715d=73d.
    Since ab=23, then a=23b=23×73d=149d.
    Thus, abcd=(14/9)d×(7/3)d(7/15)d×d=(98/27)d2(7/15)d2=9827×157=149×51=709

    Solution 2

    Suppose that d=45.
    Since cd=715, then c=715d=715×45=21.
    Since cb=15, then b=5c=5×21=105.
    Since ab=23, then a=23b=23×105=70.
    Therefore, abcd=70×10521×45=10×73×3=709.
    This solution does not show that this is the value of abcd for all such values of a, b, c, and d, but the text of the problem implies that the value is the same for any choice of a, b, c, and d that satisfy the restrictions. Solution 1 justifies that this is the value for all choices of a, b, c, and d.

    Answer: 709

  5. When three numbers a, b, c have the property that the middle number, b, is the average of the other two, then b=a+c2 tells us that 2b=a+c or b+b=a+c and so bc=ab.
    In other words, if bc=d (that is, if b=c+d for some number d), we have ab=d, which gives a=b+d=(c+d)+d=c+2d.
    We note that the common difference d could be positive or negative.
    Since Viswanathan’s three scores have this property and his score in the third game is 25, then his scores in the first two games can be written as 25+2d and 25+d, as shown in the table:

    1st game 2nd game 3rd game
    Viswanathan 25+2d 25+d 25
    Magnus

    Since Viswanathan wins either the first game or the second game, his score in this game must be at least 25. This means that d0; since the scores are all distinct, it must be the case that d>0.
    Since Viswanathan’s scores in these games are both larger than 25, Magnus’s scores in these games are either 2 more and 2 less than Viswanathan’s or 2 less and 2 more than Viswanathan’s, depending on the order in which they won these games.

    Suppose that Viswanathan wins the first game and Magnus wins the second game.
    Then Magnus’s score in the first game is 2 less than Viswanathan’s (and so is 23+2d) and Magnus’s score in the second game is 2 more than Viswanathan’s (and so is 27+d).

    1st game 2nd game 3rd game
    Viswanathan 25+2d 25+d 25
    Magnus 23+2d 27+d

    Since scores of a, b, c in order of the three games gives 2b=a+c, then c=2ba.
    Here, this means that Magnus’s score in the third game is 2(27+d)(23+2d)=31, which is not possible since a final score of 31 to 25 is not possible.

    Suppose that Magnus wins the first game and Viswanathan wins the second game.
    Then Magnus’s score in the first game is 2 more than Viswanathan’s (and so is 27+2d) and Magnus’s score in the second game is 2 less than Viswanathan’s (and so is 23+d).

    1st game 2nd game 3rd game
    Viswanathan 25+2d 25+d 25
    Magnus 27+2d 23+d

    Here, Magnus’s score in the third game is 2(23+d)(27+2d)=19, which is possible since a score of 25 to 19 is possible.
    Therefore, Magnus’s score in the third game is 19. We note that this answer does not depend on d, which tells us that this score will be 19 regardless of the value of d>0.

    Answer: 19

  6. For the product abc to be a multiple of 12, the integers a, b and c must include between them at least 1 factor of 3 and 2 factors of 2.
    Among the integers in the list, there are five categories with respect to divisibility by 2 and 3:

    We work through two possibilities: either 30 is chosen or 30 is not chosen. We do not need to worry about whether 30 is a, b or c, since the three chosen integers can be arranged in increasing order after they are chosen.

    Case 1: 30 is chosen

    In this case, at least one of the remaining integers is even, since 30 includes only 1 factor of 2.

    If both of the remaining integers are even, then 2 of the 4 integers from (B) and (C) are chosen. There are 6 ways to do this: 22 and 46; 22 and 50; 22 and 8; 46 and 50; 46 and 8; 50 and 8.
    Therefore, there are 6 ways to choose in this sub-case.

    If only one of the remaining integers is even, this even integer can come from either (B) or (C). There are 4 such integers.
    The third integer chosen must be odd, and so is from (A) or (D). There is a total of 7 integers in these.
    Thus, in this sub-case there are 4×7=28 ways to choose since each of the 4 even integers can be paired with each of the 7 odd integers.

    Case 2: 30 is not chosen

    In this case, one or two of the three integers chosen must be from (D), since abc must include at least 1 factor of 3. (If all three were from (D), there would be no factors of 2.)

    If two of the integers are from (D), there are 6 ways of choosing these 2 integers (21 and 27; 21 and 33; 21 and 39; 27 and 33; 27 and 39; 33 and 39).
    The third integer chosen must then be a multiple of 4 in order for abc to have at least 2 factors of 2. Thus, the third integer is 8.
    Thus, in this sub-case there are 6 ways to choose.

    If one of the integers is from (D), there are 4 ways of choosing that integer.
    The remaining two integers chosen then need to include a multiple of 4 and another even integer, or a multiple of 4 and an odd integer not divisible by 3, or two even integers that are not multiples of 4. (Can you see why these are all of the possible cases?)
    If these integers are a multiple of 4 and another even integer, the multiple of 4 must be 8 and the even integer comes from the 3 integers in (B).
    Thus, there are 4×3=12 ways to choose in this sub-case, since each of the 4 integers from (D) can be paired with each of the 3 integers from (B).
    If the two integers are a multiple of 4 and an odd integer not divisible by 3, the multiple of 4 must be 8 and the odd integer comes from the 3 integers in (A).
    Thus, there are 4×3=12 ways to choose in this sub-case.
    If the two integers are even and not multiples of 4, we choose 2 of the 3 integers from (B) and there are 3 ways to do this, as we saw earlier.
    Thus, there are 4×3=12 ways to choose in this sub-case.

    In total, there are 6+28+6+12+12+12=76 ways to choose the three integers.

    Answer: 76

PART B

    1. Since the sum of the entries in the left column is 29, the bottom left entry is 29139=7.
      Since the sum of the entries in the horizontal row is 29, the remaining two entries have a sum of 299=20. Since the possible entries are 3, 5 and 15, these entries must be 5 and 15.
      This means that the 3 must go in the bottom right box, which means that the middle box in the right column must contain 29113=15, which means that 5 goes in the middle box.

    2. The sums of the left and right columns are equal. Thus, 15+(t+1)+11=(2t3)+10+14t+27=2t+216=t If t=6, we complete the H to verify that the entries are all different:

      From top to bottom, the left column has the numbers 15, 7 and 11. The right column has the numbers 9, 10, and 14. The middle box has the number 16.

    3. Since the three sums are equal, we have a+12+c=12+9+7=b+7+11 and so a+c+12=28=b+18 which gives a+c=16 and b=10.
      Since a+c=16 and a and c are positive integers with a<c, we have the following possibilities:

      • a=1 and c=15; all 7 integers are different in this case

      • a=2 and c=14; all 7 integers are different in this case

      • a=3 and c=13; all 7 integers are different in this case

      • a=4 and c=12; this is not possible since there would be two 12s

      • a=5 and c=11; this is not possible since there would be two 11s

      • a=6 and c=10; this is not possible since there would be two 10s

      • a=7 and c=9; this is not possible since there would be two 7s (and two 9s)

      Thus, the possible values of a are 1, 2 and 3, giving the following figures:

      From top to bottom, the left column has 1, 12 and 15, and the right column has 10, 7, and 11. The middle box has 9.     From top to bottom, the left column has 2, 12 and 14, and the right column has 10, 7, and 11. The middle box has 9.     From top to bottom, the left column has 3, 12 and 13, and the right column has 10, 7, and 11. The middle box has 9.

    4. Since the three sums are equal, we have 7+10+k=10+n+18=(n+6)+18+4 which gives k+17=n+28=n+28, which means that k=n+11.
      Since k is between 4 and 18, inclusive, and k=n+11, then we must have n7. Since n is between 4 and 18, inclusive, then n4 as well.
      Putting this together, n can equal 4, 5, 6, or 7, which give values for k of 15, 16, 17, or 18, respectively, and values of n+6 of 10, 11, 12, or 13, respectively.
      Since the figure already contains 10 and 18, we cannot have k=18 or n+6=10, which means that n7 and n4.
      This means that the possible values of k are 16 and 17, which gives the following figures:

      From top to bottom, the left column has 7, 10 and 16, and the right column has 11, 18, and 4. The middle box has 5.     From top to bottom, the left column has 7, 10 and 17, and the right column has 12, 18, and 4. The middle box has 6.

    1. When x=0, the equation 2x+3y=12 becomes 3y=12 and so y=4. Thus, the coordinates of A are (0,4).
      When y=0, the equation 2x+3y=12 becomes 2x=12 and so x=6. Thus, the coordinates of B are (6,0).
      Thus, the vertices of AOB are A(0,4), O(0,0) and B(6,0).

      Since AOB is right-angled at O, its area equals 12×AO×BO=12×4×6=12.

    2. When x=0, the equation 6x+5y=c becomes 5y=c and so y=15c. Thus, the coordinates of D are (0,15c).
      When y=0, the equation 6x+5y=c becomes 6x=c and so x=16c. Thus, the coordinates of E are (16c,0).
      Thus, the vertices of DOE are D(0,15c), O(0,0) and E(16c,0).

      Since DOE is right-angled at O, its area equals 12×DO×EO=12×15c×16c.
      Since the area of DOE is equal to 240, we obtain 12×15c×16c=240, which gives 160c2=240.
      Thus, c2=60×240=14400. Since c>0, then c=120.

    3. When x=0, the equation (2m)x+y=4m becomes y=4m. Thus, the coordinates of P are (0,4m).
      When y=0, the equation (2m)x+y=4m becomes (2m)x=4m and so x=2. (We note that 100m<n and so neither m nor n can equal 0.) Thus, the coordinates of Q are (2,0).
      When x=0, the equation (7n)x+4y=28n becomes 4y=28n and so y=7n. Thus, the coordinates of S are (0,7n).
      When y=0, the equation (7n)x+4y=28n becomes (7n)x=28n and so x=4. Thus, the coordinates of R are (4,0).
      Since n>m and 7>4, then 7n>4m and so S is farther up the y-axis than P.

      To calculate the area of quadrilateral PQRS, we subtract the area of POQ from the area of SOR.
      Therefore, 2022=12×SO×RO12×PO×QO2022=12×7n×412×4m×22022=14n4m1011=7n2m We want to find two pairs (m,n) of positive integers with 100m<n and for which 7n2m=1011.
      When m=100, we get 7n200=1011 and so 7n=1211 which gives n=173. The pair (m,n)=(100,173) satisfies the requirements.
      From 7×1732×100=1011, we add and subtract 14 to obtain 7×173+142×10014=1011 which gives 7×173+7×22×1002×7=1011 and so 7×1752×107=1011 Therefore, (m,n)=(107,175) is a second pair that satisfies the given requirements.

    1. Solution 1

      Since 10 minutes is 16 of an hour, when Beatrice walks at 5 km/h for 10 minutes, she walks a distance of 5 km/h×16 h=56 km.
      Since Hieu cycles at 15 km/h and Beatrice walks at 5 km/h, then Hieu catches up to Beatrice at a rate of 15 km/h5 km/h=10 km/h.
      In other words, Hieu closes the 56 km gap at a rate of 10 km/h, which takes 56 km10 km/h, or 560 h, or 5 minutes.

      Solution 2

      When Beatrice walks at 5 km/h for 10 minutes, she walks a distance of 5 km/h×16 h or 56 km.
      Suppose that Hieu takes t hours to catch up to Beatrice.
      Since Hieu cycles at 15 km/h, Hieu cycles 15 km/h×t h=15t km in total over the course of t hours.
      Using similar reasoning, Beatrice walks 5t km after her initial 56 km.
      When Hieu catches up to Beatrice, they have travelled exactly the same distance, and so 15t=5t+56 which gives 10t=56 and so t=560.
      Thus, Hieu catches up to Beatrice after 560 h, which is equivalent to 5 minutes.

      1. There are 60 integers in the interval from 0 to 59 inclusive.
        This means that there are 60 possible values for b and 60 possible values for h.
        Since b and h are chosen randomly and independently, then there are 60×60=3600 possible pairs of values for b and h.
        We say that Beatrice and Hieu meet if they are at the same place on the path at the same time, including possibly starting or ending together.
        To determine the probability that Beatrice and Hieu meet, we count the number of pairs of values for b and h for which they meet, and then divide this total by 3600.
        Since the path is 2 km long and Beatrice walks at 5 km/h, then Beatrice takes 2 km5 km/h or 25 h or 24 minutes to walk the length of the path.
        Since the path is 2 km long and Hieu cycles at 15 km/h, then Hieu takes 2 km15 km/h or 215 h or 8 minutes to cycle the length of the path.
        Since Beatrice walks the path in 24 minutes and Hieu cycles the path in 8 minutes, then Beatrice and Hieu will meet on the path whenever hb is at least 0 and at most 248=16. Here is an explanation for why this is true:

        • If Hieu starts before Beatrice (that is, if h<b), since Hieu is faster than Beatrice, then Beatrice cannot catch up to Hieu, and so they never meet on the path.

        • If Hieu starts at the same time as Beatrice (that is, if h=b), then they meet at the start of the path.

        • Suppose that Hieu starts after Beatrice (that is, if h>b) but fewer than 16 minutes after Beatrice (that is, hb<16).
          We note that Hieu finishes at h+8 minutes after 9:00 a.m., and Beatrice finishes at b+24 minutes after 9:00 a.m.
          Since hb<16, then h<b+16 and so h+8<b+24, which means that Hieu finishes before Beatrice.
          Since Hieu starts after Beatrice and finishes before Beatrice, then Hieu must pass Beatrice along the path, and so they meet.

        • If Hieu starts exactly 16 minutes after Beatrice (that is, if hb=16), then Hieu and Beatrice finish at exactly the same time (because h+8=b+24) and so meet at the end of the path.

        • If Hieu starts more than 16 minutes after Beatrice (that is, if hb>16), then Hieu will not catch up to Beatrice. In this case, this is because Hieu is “behind” where Hieu would be if Hieu started exactly 16 minutes after Beatrice, and in the previous case, Hieu just caught up to Beatrice at the very end of the path.

        So we need to count the number of pairs of integer values for h and b with h and b between 0 and 59, inclusive, and with hb0 and hb16.
        To do this, we work through the values of b from 0 to 59, in each case determining the number of values of h that satisfy the restrictions.
        If b=0, then h can be any of 0 to 16, inclusive. There are 17 such values.
        If b=1, then h can be any of 1 to 17, inclusive. There are again 17 such values.
        In general, if b0 and b43, then h can be any of b to b+16 inclusive, and since b43, we have b+1659, which means that each of these 17 values satisfy the required conditions. Note that there are 44 values of b for which there are 17 values of h.
        If b=44, then h can be any of 44 to 59, inclusive. We note while h=60 is 16 greater than b=44, it is not an allowed value for h. In this case, there are 16 values for h.
        If b=45, then h can be any of 45 to 59, inclusive. There are 15 values for h.
        As b increases by 1 from 45 to 59, the number of values of h decreases by 1 in each case, since the maximum value of h does not change while its minimum value increases by 1.
        When b=59, there is 1 value of h (namely, h=59) that satisfies the conditions.
        Therefore, if N is the number of pairs of values of h and b that satisfy the given conditions, then N=44×17+16+15+14++3+2+1=44×17+(16+1)+(15+2)++(10+7)+(9+8)=44×17+8×17=52×17=884 Finally, this means that the probability that Hieu and Beatrice are at the same place on the path at the same time is N3600 or 8843600, which simplifies to 221900.

      2. As in (i), there are 3600 possible pairs of values for b and h.
        If M of these pairs result in Hieu and Beatrice meeting, then the probability that they meet is M3600.
        We are told that the probability is 13200, and so M3600=13200 which means that M is equal to 13×3600200=13×18=234.
        As in (i), Hieu takes 8 minutes to cycle the length of the path.
        Since Beatrice rides a scooter at x km/h (with x>5 and x<15), then it takes Beatrice 2 kmx km/h=2x h=120x min to travel the length of the path.
        Since x>5, then 120x<24, and so Hieu is still faster than Beatrice.

        Since x<15, then 120x>8.
        Using a similar analysis to that in (i), Hieu and Beatrice meet whenever hb is at least 0 and at most 120x8.
        This means that we need to determine the range of possible values of x for which there are 234 pairs of integer values of h and b for which each of h and b is between 0 and 59, inclusive, and hb0 and hb120x8.
        Since h and b are integers, then hb is an integer.
        Let t be the largest integer less than or equal to 120x8.
        Note that, since t is the largest such integer, t+1 is greater than 120x8.
        Note also that 120x8 is greater than 0 and less than 16, which means that t is between 0 and 15, inclusive. This means that subject to the range restrictions on h and b, the value of hb can be from 0 to t, inclusive.
        As in (i), we count the number of pairs of values for h and b by working systematically through the values of b.
        When b=0, there are t+1 possible values for h, namely the integers from 0 to t, inclusive.
        When b=1, there are t+1 possible values for h, namely the integers from 1 to t+1, inclusive.
        Similarly, while b59t, there are t+1 possible values for h. Since these values of b run from 0 to 59t, inclusive, there are 60t such values of b.
        When b=(59t)+1=60t, the largest possible value of h is still 59 (which corresponds to hb=t1) and so there are t possible values for h.
        As in (i), the number of possible values of h decreases by 1 each time b increases by 1 until b=59.
        Since the total number of pairs of values for b and h is 234, then 234=(60t)(t+1)+t+(t1)++2+1234=(60t)(t+1)+12t(t+1)468=2(60t)(t+1)+t(t+1)468=(1202t)(t+1)+t(t+1)468=(120t)(t+1) When t=3, the right side is equal to 117×4 which equals 468.
        Expanding, we obtain 468=t2+119t+120 or t2119t+348=0.
        Factoring, we obtain (t3)(t116)=0.
        Since t is between 0 and 15, inclusive, then t=3.
        Since t is 3, the greatest integer less than or equal to 120x8 is 3.
        This means that 120x83 and 120x8<4.
        Thus, 120x11 and 120x<12, which tells us that 10<x12011, which is the range of possible values for x.