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2020 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 18, 2020
(in North America and South America)

Thursday, November 19, 2020
(outside of North American and South America)

©2020 University of Waterloo


Part A

  1. We write each of the three given fractions as a decimal: 14=0.25410=0.441100=0.41 The five numbers are thus 0.25, 0.4, 0.41, 0.04, 0.404.
    Re-writing this list in increasing order, we obtain 0.04, 0.25, 0.4, 0.404, 0.41.
    Therefore, the middle number is 0.4 or 410.

    Answer: 410

  2. Each 8 by 10 rectangle has area 8×10=80.
    The 4 by 4 square has area 4×4=16.
    Each of the two shaded pieces is the part of an 8 by 10 rectangle outside a 4 by 4 square, and so has area 8016=64.
    Therefore, the total area of the shaded region is 2×64=128.
    (The area of the shaded region could also be determined by dividing it into rectangular pieces.)

    Answer: 128

  3. After 10 days, Juan has removed 1+2+3+4+5+6+7+8+9+10=55 candies.
    On day 11, Juan removes 11 candies so has now removed 55+11=66 candies.
    This means that, after day 10, Juan has removed fewer than 64 candies, and, after day 11, Juan has removed more than 64 candies.
    Therefore, the smallest possible value of n is 11.

    Answer: 11

  4. Solution 1
    The inner polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the shorter parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed between two identical trapezoids).
    The sum of the measures of the angles in this polygon with 9 sides is 180(92) or 1260.
    Thus, the measure of each of the interior angles is 1260÷9=140.
    Let θ=ABC.
    Since the nine trapezoids are identical, then the trapezoid adjacent to the left also has an angle of θ, as shown:

    Nine trapezoids forming a ring. The inner and outer edges of the ring are polygons with 9 sides. The angles surrounding one vertex of the inner polygon are theta, theta, and 140 degrees.

    Since the sum of the measures of angles around a point is 360, then 2θ+140=360 and so 2θ=220 which gives ABC=θ=110.

    Solution 2
    The outer polygon formed by the nine trapezoids is a regular polygon with 9 sides. This is because (i) all of the sides of this polygon are equal in length (equal to the longer parallel side of one of the trapezoids), and (ii) the measure of each of the interior angles of this polygon is equal (formed by two of the base angles of two identical trapezoids.)
    The sum of the measures of the angles in this polygon with 9 sides is 180(92) or 1260.
    Thus, the measure of each of the interior angles is 1260÷9=140.
    Each of these interior angles is made up from two equal base angles, which are thus equal to one-half of 140 or 70.
    Since the trapezoids have parallel sides, then ABC is supplementary to one of these base angles, and so ABC=18070=110.

    Answer: 110

  5. Throughout this solution, we use the fact that if a and b are positive, then 1a is less than 1b exactly when a is greater than b. (That is, 1a<1b exactly when a>b.)
    Since 1x+1y is positive, then x and y cannot both be negative (otherwise 1x+1y would be negative).
    Since xy, this means that either both x and y are positive or x is negative and y is positive.
    Suppose that x and y are both positive. This means that 1x and 1y are both positive.
    Since 1x+1y=14, then 1x and 1y are both less than 14.
    This means that x>4 and y>4.
    Since xy, then 1x1y which means that 14=1x+1y1x+1x or 142x which gives 282x and so x8. (In other words, since 1x is larger, then it must be at least one-half of 14 so 1x is at least 18 and so x is at most 8.)
    Since x>4 and x8, then x could equal 5, 6, 7, or 8.
    If x=5, then 1y=1415=520420=120 and so y=20.
    If x=6, then 1y=1416=312212=112 and so y=12.
    If x=7, then 1y=1417=728428=328 and so y=283 which is not an integer.
    If x=8, then 1y=1418=2818=18 and so y=8.
    Therefore, when x and y are both positive, the solutions are (x,y)=(5,20),(6,12),(8,8).
    Suppose that x is negative and y is positive. This means that 1x is negative and 1y is positive.
    Since 1x+1y=14, then 1y is greater than 14.
    This means that y<4.
    Since y is positive and y<4, then y could equal 1, 2 or 3.
    If y=1, then 1x=1411=1444=34 and so x=43, which is not an integer.
    If y=2, then 1x=1412=1424=14 and so x=4.
    If y=3, then 1x=1413=312412=112 and so x=12.
    Therefore, when x is negative and y is positive, the solutions are (x,y)=(4,2),(12,3).
    In summary, the solutions are (x,y)=(4,2),(12,3),(5,20),(6,12),(8,8).

    Answer: (x,y)=(4,2),(12,3),(5,20),(6,12),(8,8)

  6. When each of 90 players plays against all of the other 89 players exactly once, the total number of games played is 90×892=4005.
    This is because each of the 90 players plays 89 games, giving 90×89, but each of the games is counted twice in this total, since each game is played by two players, giving 90×892.
    Exactly 1 point is awarded in total for each game. This is because the two players either win and lose (1+0 points) or both tie (0.5+0.5 points).
    Therefore, over 4005 games, a total of 4005 points are awarded to the players in the league.
    Suppose that, after all games have been played, n players have at least 54 points.
    Then 54n4005 since these n players cannot have more points than the total number of points awarded to the entire league.
    Since 40055474.17 and n is an integer, this means that n74. (Note that 74×54=3996 and 75×54=4050.)
    Suppose that 74 players have at least 54 points.
    This accounts for at least 74×54=3996 of the total of 4005 points.
    This leaves at most 40053996=9 points to be distributed among the remaining 9074=16 players.
    These 16 players play 16×152=120 games amongst themselves (each of 16 players plays the other 15 players) and these games generate 120 points to be distributed among the 16 players.
    This means that these 16 players have at least 120 points between them, and possibly more if they obtain points when playing the other 54 players.
    Therefore, these 16 players have at most 9 points and at least 120 points in total, which cannot happen.
    Thus, it cannot be the case that 74 players have at least 54 points.
    Suppose that 73 players have at least 54 points.
    This accounts for at least 73×54=3942 of the total of 4005 points.
    This leaves at most 40053942=63 points to be distributed among the remaining 9073=17 players.
    These 17 players play 17×162=136 games amongst themselves.
    This means that these 17 players have at least 136 points between them.
    Therefore, these 17 players have at most 63 points and at least 136 points in total, which cannot happen.
    Thus, it cannot be the case that 73 players have at least 54 points.
    Suppose that 72 players have at least 54 points.
    This accounts for at least 72×54=3888 of the total of 4005 points.
    This leaves at most 40053888=117 points to be distributed among the remaining 9072=18 players.
    These 18 players play 18×172=153 games amongst themselves.
    This means that these 18 players have at least 153 points between them.
    Therefore, these 18 players have at most 117 points and at least 153 points in total, which cannot happen.
    Thus, it cannot be the case that 72 players have at least 54 points.
    Suppose that 71 players have at least 54 points.
    This accounts for at least 71×54=3834 of the total of 4005 points.
    This leaves at most 40053834=171 points to be distributed among the remaining 9071=19 players.
    These 19 players play 19×182=171 games amongst themselves.
    This means that these 19 players have at least 171 points between them.
    These point totals match. Therefore, it appears that 71 players could have at least 54 points.
    It is in fact possible for this to happen.
    Suppose that every game between two of the 71 players ends in a tie, every game between one of the 71 players and one of the 19 players ends in a win for the first player, and every game between two of the 19 players ends in a tie.
    In this case, each of the 71 players has 70 ties and 19 wins, for a point total of (70×0.5)+(19×1)=54, as required.

    Answer: 71

Part B

    1. In the arithemetic sequence 20,13,6,1, each term is obtained by adding a constant.
      This constant is 1320=7.
      Therefore, the next two terms are 1+(7)=8 and 8+(7)=15.

    2. The arithmetic sequence 2,a,b,c,14 has 5 terms.
      This means that the constant difference (that is, the constant that is added to one term to obtain the next) is added 4 times to the first term (which equals 2) to obtain the last term (which equals 14).
      Thus, the constant difference must be 1424=124=3.
      This means that the sequence starts at 2 and counts by 3s: 2, 5, 8, 11, 14.
      Therefore, a=5, b=8, and c=11.

    3. There are six possible ways in which the three terms can be arranged: 7,15,t15,7,t7,t,1515,t,7t,7,15t,15,7 We determine the corresponding value of t in each case:

      • 7,15,t: Here, the constant difference is 157=8. Thus, t=15+8=23.

      • 15,7,t: Here, the constant difference is 715=8. Thus, t=7+(8)=1.

      • t,7,15: Here, the constant difference is 157=8. Thus, t+8=7 and so t=1.

      • t,15,7: Here, the constant difference is 715=8. Thus, t+(8)=15 and so t=23.

      • 7,t,15: Here, the constant difference is added 2 times to 7 to get 15. Thus, the constant difference is 1572=82=4. Thus, t=7+4=11.

      • 15,t,7: Here, the constant difference is 7152=82=4. Thus, t=15+(4)=11.

      Therefore, the possible values of t are 23, 1, and 11.

    4. Suppose that the sequence r,s,w,x,y,z has constant difference d.
      This means that the difference between any two consecutive terms is d.
      Since the sequence has 6 terms, then the difference is added 5 times to r to get z. In other words, zr is equal to 5d.
      Also, the difference between a later term in the sequence and an earlier term in the sequence must equal one of d, 2d, 3d, 4d, or 5d, depending on the number of terms between the two specific terms.
      Since 4 and 20 are two terms in the sequence, with 20 appearing later than 4, and 204=16, then either d=16 or 2d=16 (which gives d=8) or 3d=16 (which gives d=163) or 4d=16 (which gives d=4) or 5d=16 (which gives d=165).
      The corresponding values of 5d are 80, 40, 803, 20, and 16.
      Since zr is equal to 5d, then the largest possible value of zr is 80 and the smallest possible value is 16.
      (The sequence 4,20,36,52,68,84 shows that a difference of 80 is possible.
      The sequence 4,365,525,685,845,20 shows that a difference of 16 is possible.)

      1. The volume of Tank B is 5 cm×9 cm×8 cm=360 cm3.
        This means that Tank B is 13 full when it contains 13×360 cm3=120 cm3 of water.
        Since Tank B fills at 4 cm3/s, then Tank B is 13 full after 120 cm34 cm3/s=30 s.

      2. Since the volume of Tank B is 360 cm3 and it fills at 4 cm3/s, then it is full after 360 cm34 cm3/s=90 s.
        (We can also note that since it is 13 full after 30 s and it fills at a constant rate, then it is completely full after 3×30 s=90 s.)
        Since Tank A drains at 4 cm3/s, then in 90 s, 360 cm3 of water has drained out.
        Since the volume of Tank A is 10 cm×8 cm×6 cm=480 cm3, then after 90 s, the volume of water in Tank A is 480 cm3360 cm3=120 cm3.
        Tank A is sitting on one of its 10 cm×8 cm faces, which has an area of 80 cm2.
        Since the water itself forms a rectangular prism, the depth of water in Tank A at this instant is 120 cm380 cm2=1.5 cm.

      3. Since Tank A starts full and Tank B starts empty and the rate at which water leaves Tank A equals the rate at which water enters Tank B, then the combined volume of water in the two tanks is constant and equal to the initial volume of Tank A, or 480 cm3.
        Let the depth of water in Tank A and Tank B when their depths are equal be d cm.
        Since the area of the bottom face of Tank A is 80 cm2, then the volume of water in Tank A when the depth is d cm is 80d cm3.
        Since the area of the bottom face of Tank B is 5 cm×9 cm=45 cm2, then the volume of water in Tank B when the depth is d cm is 45d cm3.
        Since the combined volume is 480 cm3, then 80d+45d=480 which gives 125d=480 or 25d=96 and so d=9625=3.84.
        Thus, the depth is 3.84 cm.

    1. Suppose that the volumes of water in Tank C and Tank D are equal t seconds after Tank D begins to fill.
      The volume of Tank C is 31 cm×4 cm×4 cm=496 cm3.
      Since Tank C starts full, it drains at 2 cm3/s, and will have drained for t2 seconds, then the volume of water in Tank C after t seconds is (4962(t2)) cm3.
      Since Tank D starts empty and fills at 1 cm3/s, then the volume of water in Tank D after t seconds is t cm3.
      Since these volumes are equal, then 4962(t2)=t4962t+4=t500=3tt=5003 Therefore, the volumes are equal after 5003 seconds. At this time, the volume of water in Tank D is 5003 cm3.
      Suppose that the depth of water in Tank D at this instant is d cm. Note that the water forms a square-based pyramid with its top pointing downwards. The surface of the water in Tank D will be a square with side length 2d cm.

      Tank D is an inverted square-based pyramid with height 10 cm and base side length 20 cm. The tank is partially filled with water up to a depth of d cm.

      This is because, as Tank D fills, the ratio of the side length of the base to the height will remain constant. This ratio for the whole tank is 20 cm:10 cm which is equivalent to 2:1. We could say that the square-based pyramid formed by the water at any instant is similar to the Tank itself.
      The volume of a square-based pyramid whose base has an edge length of 2d cm and whose height is d cm is 13(2d)2d cm3, which equals 43d3 cm3.
      Therefore, 43d3=5003 which gives d3=125 and so d=5.
      Therefore, at the instant when the volume of water in Tank C equals the volume of water in Tank D, the depth of water in Tank D is 5 cm.

    1. P can be a multiple of 216.
      For example, if a=6, b=6, c=6, and d=2, then P=abcd=432 which equals 2×216.

    2. P cannot be a multiple of 2000.
      To see this, we note first that 2000=2×10×10×10=24×53.
      For P=abcd to be a multiple 2000, P would need to include at least 4 factors of 2 and at least 3 factors of 5.
      Thus the result of the product abcd would need to include at least 3 factors of 5.
      Since the only values for a,b,c,d are 1,2,3,4,5,6,7,8,9, the only way to introduce factors of 5 are for one or more of the integers a,b,c,d to equal 5. There are no other multiples of 5 in this list.
      To be a multiple of 2000, P must have 3 factors of 5, so at least 3 of a,b,c,d equal 5.
      Suppose that a=b=c=5. Then P=5×5×5×d=125d.
      In this case, P is at most 125×9=1125 which is too small to be a multiple of 2000.
      Alternatively, we could note that d cannot include the required 4 factors of 2 since there are no multiples of 24=16 in the list.
      Therefore, P cannot be a multiple of 16 and thus cannot be a multiple of 2000.

    3. We note first that 27=128 and 210=1024.
      For P to be divisble by 27=128 but not by 210=1024, the integer P needs to have at least 7 factors of 2 but fewer than 10 factors of 2. In other words, P must be divisible by 27, could be divisible by 28 or by 29, but cannot be divisible by any larger power of 2.
      Assume that a has A factors of 2, b has B factors of 2, c has C factors of 2, and d has D factors of 2. Therefore, 7A+B+C+D9.
      We note that we need to count possible values of P, not quadruples (a,b,c,d).
      As a result, we may assume that ABCD. In other words, a has at least as many factors of 2 as b, b has at least as many factors of 2 as c, and so on.
      Among the possible values of a, b, c, d, there is one that includes exactly 3 factors of 2 (namely, 8), and one that includes exactly 2 factors of 2 (namely, 4), and two that include exactly 1 factor of 2 (namely, 2 and 6).
      Therefore, each of A, B, C, D is at least 0 and at most 3.
      Suppose A+B+C+D=9. Since 3ABCD0, then we have the following possibilities: 3+3+3+0=93+3+2+1=93+2+2+2=9 In each case, we determine the possible values of P:

      • 3+3+3+0=9: Here, a=b=c=8 and d is one of 1, 3, 5, 7, or 9, which gives 5 possible values of P: 29, 3×29, 5×29, 7×29, 9×29.

      • 3+3+2+1=9: Here, a=b=8 and c=4 and d=2 or d=6. There are no additional values of P here, as we get 29 and 3×29 as possible values for P.

      • 3+2+2+2=9: Here, a=8 and b=c=d=4. There are no additional values of P here, as the only possible value of P here is 29.

      Suppose A+B+C+D=8. Since 3ABCD0, then we have the following possibilities: 3+3+2+0=83+3+1+1=83+2+2+1=82+2+2+2=8 In each case, we determine the possible values of P:

      • 3+3+2+0=8: Here, a=b=8 and c=4 and d is one of 1, 3, 5, 7, or 9, which gives 5 additional values of P: 28, 3×28, 5×28, 7×28, 9×28.

      • 3+3+1+1=8: Here, a=b=8 and c and d each equal 2 or 6. This means that P can equal 28 or 3×28 or 9×28, which give no additional values of P.

      • 3+2+2+1=8: Here, a=8 and b=c=4 and d=2 or d=6. There are no additional values of P here.

      • 2+2+2+2=8: Here, a=b=c=d=4. There are no additional values of P here.

      Suppose A+B+C+D=7. Since 3ABCD0, then we have the following possibilities: 3+3+1+0=73+2+2+0=73+2+1+1=72+2+2+1=7 In each case, we determine the possible values of P:

      • 3+3+1+0=7: Here, a=b=8 and c=2 or c=6 and d is one of 1, 3, 5, 7, or 9. From these, we get eight additional values of P: 27, 3×27, 5×27, 7×27, 9×27, 15×27, 21×27, 27×27.

      • 3+2+2+0=7: Here, a=8 and b=c=4 and d=1,3,5,7,9. There are no additional values of P here.

      • 3+2+1+1=7: Here, a=8 and b=4 and each of c and d is 2 or 6. There are no additional values of P here.

      • 2+2+2+1=7: Here, a=b=c=4 and d is 2 or 6. There are no additional values of P here.

      In total, there are 5+5+8=18 possible values of P.

    4. Since P is 98 less than a multiple of 100, then P=2 or the last two digits of P are 02.
      A positive integer is divisible by 5 exactly when its units digit is either 0 or 5. Therefore, P is not divisible by 5.
      A positive integer is divisible by 4 exactly when the integer formed by its last two digits is divisible by 4. Since 2 is not divisible by 4, P is not divisible by 4.
      Since P=abcd and each of a, b, c, d is equal to one of 1, 2, 3, 4, 5, 6, 7, 8, 9, this means that

      • none of a, b, c, d equals 5 because P is not a multiple of 5, and

      • none of a, b, c, d equals 4 or 8 because P is not a multiple of 4, and

      • exactly one of a, b, c, d equals 2 or 6 because P is even but not a multiple of 4.

      In other words, exactly one of a, b, c, d equals 2 or 6 and the remaining variables are restricted to being equal to 1, 3, 7, or 9.
      For the moment, we suppose that a equals 2 or 6 and that bcd. We will eventually lift these restrictions in order to be able to arrange the possible values into ordered quadruples.
      Suppose that a=2.
      Since P equals 2 or ends in 02, this means that P=100k+2 for some non-negative integer k.
      Since P=abcd and a=2, then 2×bcd=100k+2 or bcd=50k+1.
      Thus, bcd is 1 more than a multiple of 50, which means that bcd=1 or the final two digits of bcd are 01 or 51.
      Suppose that a=6. Here, P=6bcd. Note also that P>2 in this case which means that P102.
      Since P has a units digit of 2 and P=6×bcd, the units digit of bcd must be 2 or 7.
      We write bcd=10q+2 or bcd=10q+7 for some non-negative integer q.
      Thus, 6×bcd=60q+12 or 6×bcd=60q+42 and so 60q=P12 or 60q=P42.
      Since the final two digits of P are 02, the final two digits of 60q are either 90 or 60.
      Since 60 is divisible by 4, then 60q is divisible by 4. Since an integer ending in 90 is not divisible by 4, this case is not possible and so we cannot have bcd=10q+2.
      If the final two digits of 60q=10×(6q) are 60, then 6q has a units digit of 6, and so q must have a units digit of 1 or 6. (We can see this by testing all of the possible units digits of q.)
      Since bcd=10q+7, then the final two digits of bcd are 17 or 67.
      We now need to determine what, if any, products bcd have final two digits 01, 51, 17, 67, where bcd, each of b, c, d equals one of 1, 3, 7, 9, and a=2 (digits 01 or 51) or a=6 (digits 17 or 67).
      Suppose that b=c=1 and so bcd=d. The only possibility that gives the correct digits is d=1, which corresponds to a=2.
      Suppose that b=1 and c>1. Then bcd=cd. Here, we have the following possibilities for cd: 3×3=93×7=213×9=277×7=497×9=639×9=81 We do not get the correct digits here.
      Suppose that b=3 and so bcd=3×cd. Since 3cd, we can use the possibilities from the previous case. Here, we can multiply the previous possibilities by 3 to see that bcd can be equal to any of 27, 63, 81, 147, 189, 243, none of which has the correct digits.
      Suppose that b=7 and so 7cd. The possibilities here are: 7×7×7=3437×7×9=4417×9×9=567 The last case gives the correct digits for P when a=6.
      Finally, if b=9, then we must have b=c=d=9 and so bcd=729, which does not end with the correct digits.
      Therefore, we could have a=2 with b=c=d=1 or a=6 with b=7 and c=d=9.
      Finally, we arrange 2,1,1,1 and 6,7,9,9 into ordered quadruples.
      In the first case, there are 4 spots into which one can place the 2, and so there are 4 ordered quadruples in this case.
      In the second case, there are 4 spots into which to first put the 6 and then 3 remaining spots into which to put the 7, leaving the 9s to be put without choice into the remaining spots. This gives 4×3=12 ordered quadruples.
      In total, there are 4+12=16 ordered quadruples (a,b,c,d) for which the product P is 98 less than a multiple of 100.