CEMC Banner

2019 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 20, 2019
(in North America and South America)

Thursday, November 21, 2019
(outside of North American and South America)

©2019 University of Waterloo


PART A

  1. Since ABC is equilateral, then ABC=60. (ABC has three equal angles whose measures add to 180.)
    Since BDC is right-angled at D and has DB=DC, then BDC is a right-angled isosceles triangle, which makes DBC=45. (Here, DBC=DCB since DB=DC and the measures of the two angles add to 90, which makes each 45.)
    Therefore, x=ABD=ABCDBC=6045=15.
    Thus, x=15.

    Answer: x=15

  2. Solution 1
    Binh’s 20 quarters are worth 20×25=500 cents.
    Abdul’s 20 dimes are worth 20×10=200 cents.
    Since Binh’s and Abdul’s coins have the same total value, then the value of Abdul’s quarters is 500200=300 cents.
    Since each quarter is worth 25 cents, then Abdul has 300÷25=12 quarters.

    Solution 2
    Binh’s 20 quarters are worth 20×25=500 cents.
    Abdul’s 20 dimes are worth 20×10=200 cents.
    Suppose that Abdul has x quarters. These are worth 25x cents.
    Since Binh’s and Abdul’s coins have the same total value, then 500=200+25x and so 25x=500200=300 and so x=30025=12.

    Answer: 12 quarters

  3. We note that 36000=36×1000=62×103=(2×3)2×(2×5)3=22×32×23×53=25×32×53 This is called the prime factorization of 36 000. There are many different ways of getting to this factorization, although the final answer will always be the same.
    Since 36000=25×32×53 and we want 36000=2a3b5c, then a=5 and b=2 and c=3.
    Thus, 3a+4b+6c=3×5+4×2+6×3=15+8+18=41.

    Answer: 41

  4. Ali earns a total of 12 points for her 12 correct answers. To determine her possible total scores, we need to determine her possible numbers of bonus points.
    Since Ali answers 3 questions incorrectly, these could all be in 1 category, split over 2 categories (2 from 1 category and 1 from another), or split over 3 categories (1 from each).
    In the first case, she answers all of the questions in 4 of the 5 categories correctly, and so earns 4 bonus points. In this case, her total score would be 12+4=16.
    In the second case, she answers all of the questions in 3 of the 5 categories correctly, and so earns 3 bonus points. In this case, her total score would be 12+3=15.
    In the third case, she answers all of the questions in 2 of the 5 categories correctly, and so earns 2 bonus points. In this case, her total score would be 12+2=14.
    These are all of the possibilities.
    Therefore, Ali’s possible total scores are 14, 15 and 16.

    Answer: 14,15,16

  5. Since |a| is at least 0 and |b| is at least 0 and |a|+|b|10, then |a| is at most 10 and |b| is at most 10.
    We count the number of possible pairs (a,b) by working through the possible values of |a| from 0 to 10.
    Suppose that |a|=0. This means that a=0. There is 1 possible value of a in this case.
    Since |a|=0 and |a|+|b|10, then |b|10 which means that the possible values for b are 10, 9, 8, , 1, 0, 1, , 8, 9, or 10. There are 21 possible values of b in this case.
    Since there is 1 possible value for a and there are 21 possible values for b, then overall there are 1×21=21 pairs (a,b) when |a|=0.
    Suppose that |a|=1. This means that a=1 or a=1. There are 2 possible values of a in this case.
    Since |a|=1 and |a|+|b|10, then |b|9 which means that the possible values of b are 9, 8, 7, , 1, 0, 1, , 7, 8, or 9. There are 19 possible values of b in this case.
    Since there are 2 possible values for a and 19 possible values for b, then overall there are 2×19=38 pairs (a,b) when |a|=1.
    Suppose that |a|=2. This means that a=2 or a=2. There are 2 possible values of a in this case.
    Here, |b|8 which means that b could equal 8, 7, 6, , 1, 0, 1, , 6, 7, or 8. There are 17 possible values of b in this case.
    Overall, there are 2×17=34 pairs (a,b) when |a|=2.
    As |a| increases from 2 to 9, at each step, the largest possible value of |b| will decrease by 1, which means that there will be 2 fewer possible values of b from each step to the next. Since there are 2 possible values for a at each step, this means that there will be 2×2=4 fewer pairs (a,b) at each step.
    We check the final case |a|=10 to verify that nothing different happens in the last case.
    Suppose that |a|=10. This means that a=10 or a=10. There are 2 possible values of a in this case.
    Here, |b|0 which means that b can only equal 0. There is 1 possible value of b in this case.
    Overall, there are 2×1=2 pairs (a,b) when |a|=10.
    In total, this means that there are 21+38+34+30+26+22+18+14+10+6+2 pairs (a,b) with |a|+|b|10.
    Grouping the last 10 numbers in pairs from the outside towards the middle we obtain 21+(38+2)+(34+6)+(30+10)+(26+14)+(22+18) which equals 21+5×40 or 221.
    Thus, there are 221 pairs.
    (This problem can also be solved using a neat result called Pick’s Theorem. We encourage you to look this up and think about how you might apply it here.)

    Answer: 221

  6. Suppose that the original circle has radius R.
    Thus, the circumference of this circle is 2πR.
    When this circle is cut into two pieces and each piece is curled to make a cone, the ratio of the lateral surface areas of these cones is 2:1.
    This means that the ratio of the areas of the two pieces into which the circle is cut is 2:1 since it is these pieces that become the lateral surfaces of the cones.
    In other words, the sector cut out is 13 of the area of the circle, which means that its central angle is 13 of the total angle around the centre, or 13×360=120.
    Since the central angles of the two pieces are 240 and 120, which are in the ratio 2:1, then the circumference is split in the ratio 2:1 when the circle is cut.
    Since the circumference of the original circle is 2πR, then the lengths of the pieces of the circumference are 43πR and 23πR.
    These pieces become the circumferences of the circular bases of the two cones.
    Since the ratio of the circumference to the radius of a circle is 2π:1, then the radii of the bases of the two cones are 43πR2π=23R and 23πR2π=13R.
    Since the radius of the original circle becomes the slant height of each cone, then the slant height in each cone is R.
    In a cone, the radius and the height are perpendicular forming a right-angled triangle with the slant height as its hypotenuse.
    Therefore, the height of a cone with slant height R and radius 23R is R2(23R)2=R249R2=59R2=53R Also, the height of a cone with slant height R and radius 13R is R2(13R)2=R219R2=89R2=83R The volume of a cone with radius 23R and height 53R is 13π(23R)2(53R) which equals 4581πR3.
    The volume of a cone with radius 13R and height 83R is 13π(13R)2(83R) which equals 881πR3.
    Dividing the first volume by the second, we obtain 4581πR3881πR3=458=4522=4524=52=10 Therefore, the ratio of the larger volume to the smaller volume is 10:1.

    Answer: 10:1

PART B

    1. In the diagram, AC is the hypotenuse of a right-angled triangle with legs of length 9 and 12.
      By the Pythagorean Theorem, AC2=92+122=81+144=225.
      Since AC>0, then AC=225=15.
      In the diagram, CB is the hypotenuse of a right-angled triangle with legs of length 3 and 4.
      By the Pythagorean Theorem, CB2=32+42=9+16=25.
      Since BC>0, then CB=25=5.
      (Note that AC:CB=15:5=3:1.)

    2. Using the given fact, we can compute the ratio by computing the ratio of the differences of x-coordinates. We see that 11551=64=32.
      We could also have this with y-coordinates: 2557=32=32.
      Therefore, the ratio of lengths GJ:JH equals 3:2.

    3. Solution 1
      The difference between the x-coordinates of D(1,6) and E(7,9) is 71=6.
      This difference is split in the ratio 1:2 when it is written as 2+4.
      The difference between the y-coordinates of D(1,6) and E(7,9) is 96=3.
      This difference is split in the ratio 1:2 when it is written as 1+2.
      Since D has coordinates (1,6) and the x- and y-coordinates of E are larger than those of F, then F should have coordinates (1+2,6+1) or (3,7).
      Verifying, using the points D(1,6), F(3,7), E(7,9), we see that 3173=24=12and7697=12 Thus, F(3,7) splits the line segment joining D(1,6) and E(7,9) in the ratio 1:2.

      Solution 2
      Suppose that F has coordinates (a,b).
      Since F(a,b) splits D(1,6) and E(7,9) in the ratio 1:2, then a17a=12 and b69b=12.
      From a17a=12, we obtain 2a2=7a and so 3a=9 or a=3.
      From b69b=12, we obtain 2b12=9b and so 3b=21 or b=7.
      Thus, F(3,7) splits the line segment joining D(1,6) and E(7,9) in the ratio 1:2.

    4. For M(7,5) to divide K(1,q) and L(p,9) in the ratio 3:4, we need 71p7=34and5q95=34 Since 71p7=6p7 and 34=68, then 6p7=68 gives p7=8 and so p=15.
      Since 5q95=5q4, then 5q4=34 gives 5q=3 and so q=2.
      Therefore, the point M(7,5) divides the line segment joining K(1,2) and L(15,9) in the ratio 3:4.

    1. By definition,  7327 =73+27+72+37=209.

    2. We first note that a two-digit number “mn” is equal to 10m+n.
      This means that  abcd =ab+cd+ac+bd=(10a+b)+(10c+d)+(10a+c)+(10b+d)=20a+11b+11c+2d Therefore,  5bc7 =20×5+11b+11c+2×7=11b+11c+114 and  xb+1c3y =20x+11(b+1)+11(c3)+2y=20x+11b+11c+2y22 For these to be equal, we need 20x+2y22=114 and so 20x+2y=136 which means that 10x+y=68.
      Since x and y are digits, then it must be the case that x=6 and y=8. (There are no other possibilities, since x cannot be 7 or greater (since y is at least 0) and x cannot be 5 or smaller (since y is at most 9).)

    3. Using our work so far,  abcd  a+1b2c1d+1 =(20a+11b+11c+2d)(20(a+1)+11(b2)+11(c1)+2(d+1))=(20a+11b+11c+2d)(20a+11b+11c+2d+202211+2)=20+22+112=11 Therefore, the only possible value for the difference is 11.

    4. From our earlier work,  abcd =20a+11b+11c+2d.
      This means that we want to find all non-zero digits a,b,c,d for which 20a+11b+11c+2d=104 Since each of b, c and d is at least 1, then 11b+11c+2d is at least 24, which means that 20a=104(11b+11c+2d) is at most 80.
      Since 20a is at most 80, then a is at most 4.
      Since a is at least 1, then a could equal 1, 2, 3, or 4.
      Case 1: a=1
      Here, 11b+11c+2d=10420×1=84.
      Note that 11(b+c)=11b+11c=842d which is even since 822d=2(42d).
      Since 11(b+c) is even, then b+c is even.
      Since 2d is positive, then 11(b+c) is less than 84.
      This means that b+c is less than 8.
      Since d is at most 9, then 2d is at most 18, which means that 11(b+c)=842d is at least 8418=66.
      This means that b+c is at least 6.
      Since b+c is even, at least 6, and less than 8, then the only possibility is b+c=6.
      If b+c=6, then 11(b+c)=66 and so 2d=8411(b+c)=18 which means that d=9.
      Also, if b+c=6, then since b and c are non-zero digits, we could have b and c equal to 1 and 5, or 2 and 4, or 3 and 3, or 4 and 2, or 5 and 1.
      There are thus 5 grids in this case: 1159 ,  1249 ,  1339 ,  1429 ,  1519 Case 2: a=2
      Here, 11b+11c+2d=10420×2=64.
      As in Case 1, b+c is even.
      Since 2d is positive, then 11(b+c) is less than 64.
      This means that b+c is less than 6.
      Since d is at most 9, then 2d is at most 18, which means that 11(b+c)=642d is at least 6418=46.
      This means that b+c is greater than 4.
      Therefore, b+c must be an even integer, must be less than 6, and must be greater than 4.
      No such integer exists, and so there are no solutions in this case.
      Case 3: a=3
      Here, 11b+11c+2d=10420×3=44.
      As in Case 1, b+c is even.
      Since 2d is positive, then 11(b+c) is less than 44.
      This means that b+c is less than 4.
      Since d is at most 9, then 2d is at most 18, which means that 11(b+c)=442d is at least 4418=26.
      This means that b+c is greater than 2.
      Therefore, b+c must be an even integer, must be less than 4, and must be greater than 2.
      No such integer exists, and so there are no solutions in this case.
      Case 4: a=4
      Here, 11b+11c+2d=10420×4=24.
      Since b1 and c1 and d1, then 11b+11c+2d24.
      To have 11b+11c+2d, we must have b=c=d=1.
      There is thus 1 grid in this case: 4111 .
      In summary, the grids that satisfy  abcd =104 are abcd=1159,1249,1339,1429,1519,4111

    1. After 4 people, there is 1 person at the Left table, 1 person at the Centre table, and 2 people at the Right table.
      Continuing the table, we obtain

      Left Middle Right
      5 3 6 P1
      5 P2 3 3
      52 3 P3 3
      52 32 3 P4
      52 P5 32 2
      53 32 2 P6
      53 P7 32 32

      Person 5 sits at the Left table because 52=2.5 is greater than 32=1.5.
      Person 6 sits at the Right table because 531.67 and 32=1.5 are both less than 2.
      Person 7 sits at the Left table because 531.67 is greater than 32=1.5.
      Therefore, Person 5 sits at the Left table, Person 6 at the Right table, and Person 7 at the Left table.

    2. Suppose that there are integers L, M, R for which the first 6 people sit as described.
      We construct a similar chart from the given information, using the information about where the previous people have sat to calculate the shares in each row:

      Left Middle Right
      L P1 M R
      12L M P2 R
      12L 12M R P3
      12L P4 12M 12R
      13L P5 12M 12R
      14L P6 12M 12R

      Since Person 1 sits at the Left table, then their potential share at the Left table is at least as large as it is at the Middle and Right tables.
      Therefore, LM and LR.
      Since Person 2 sits at the Middle table, then their potential share at the Middle table is larger than that at the Left table (if they were equal, they would choose Left) and at least as large as it is at the Right table.
      Therefore, M>12L and MR.
      Since Person 6 sits at the Left table, then their potential share at the Left table is at least as large as it is at the Middle and Right tables.
      Therefore, 14L12M and 14L12R.
      The inequality 14L12M is equivalent to saying 12LM.
      But this gives M>12L and 12LM.
      These two inequalities cannot both be true.
      This means that we have a contradiction and so there are no values of L, M, R that would give the seating pattern shown.

    3. Since L=9, the 9th person to sit at the Left table would be getting a share of 1 kg.
      Since M=19, the 19th person to sit at the Middle table would be getting a share of 1 kg.
      Since R=25, the 25th person to sit at the Right table would be getting a share of 1 kg.
      We note also that each additional person who sits at a table causes the share per person at that table to decrease.
      Since each person sits at the table that maximizes their current share of chocolate, then there cannot be 10 people at the Left table before there are 19 people at the Middle table and 25 people at the Right table, because the 10th person at the Left table would get a larger share by sitting at the Middle or Right table.
      Similarly, there cannot be 20 people at the Middle table before there are 9 people at the Left table and 25 people at the Right table, and there cannot be 26 at the Right table before there are 9 people at the Left table and 19 people at the Middle table.
      In other words, we must have 9 people at the Left table, 19 people at the Middle table, and 25 people at the Right table before there are more than 9, 19 and 25 people at the Left, Middle and Right tables, respectively.
      At this point, there are 9+19+25=53 people in total that have been seated.
      The largest multiple of 53 less than 2019 is 38×53=2014.
      We note also that 38×9=342 and 38×19=722 and 38×25=950.
      The 342nd person to sit at the Left table would be getting a share of 9342 kg=138 kg.
      The 722nd person to sit at the Middle table would be getting a share of 19722 kg=138 kg.
      The 950th person to sit at the Right table would be getting a share of 25950 kg=138 kg.
      Using a similar argument to that above, we cannot have more than 342 people at the Left table before there are 722 people and 950 people at the Middle and Right tables. Similarly, we cannot have more than 722 people at the Middle table before there are 342 people and 950 people at the Left and Right tables, or more than 950 people at the Right table before there are 342 people and 722 people at the Left and Middle tables.
      In other words, once 2014 people have been seated, there are 342 people at the Left table, 722 people at the Middle table, and 950 people at the Right table.
      To determine at which table Person 2019 sits, we first determine where Person 2015, Person 2016, Person 2017, and Person 2018 sit:

      Left Middle Right
      93430.026239 197230.026279 259510.026288 P2015
      93430.026239 197230.026279 P2016 259520.026261
      93430.026239 197240.026243 259520.026261 P2017
      93430.026239 197240.026243 P2018 259530.026233
      93430.026239 P2019 197250.026207 259530.026233

      Therefore, Person 2019 sits at the left table.