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2022 Canadian Senior
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 2222=16, then 24=16 and so r=4.
    Since 55=25, then 52=25 and so s=2.
    Therefore, r+s=4+2=6.

    Answer: 6

  2. Solution 1

    Since x+y2=5, then x+y=25=10. Since xy2=2, then xy=22=4.

    Therefore, x2y2=(x+y)(xy)=104=40.

    Solution 2

    Since x+y2=5 and xy2=2, then x+y2+xy2=5+2 which simplifies to x=7.
    Also, x+y2xy2=52 which simplifies to y=3.
    Therefore, x2y2=7232=499=40.

    Answer: 40

  3. Suppose that the two integers are m and n.
    We are told that m+n=60 and lcm(m,n)=273.
    We note that 273=391=3713.
    Since 273 is a multiple of each of m and n, this means that m and n are both divisors of 273.
    The divisors of 273 are 1, 3, 7, 13, 21, 39, 91, 273.
    Of these, the only pair that has a sum of 60 is 21 and 39.
    Thus, the two integers are 21 and 39.
    We note that since 21=37 and 39=313, then an integer is a common multiple of 21 and 39 exactly when it has prime factors of 3, 7 and 13, so the smallest such integer is 3713=273 which makes lcm(21,39)=273.

    Answer: 21 and 39

  4. By the Pythagorean Theorem, CD2=AC2AD2=62526002=390625360000=30625=1752 Since CD>0, we get CD=175.
    Let DAC=θ. Thus, BAC=2DAC=2θ.

    Looking in ADC, we see that sinθ=CDAC=175625=725 and cosθ=ADAC=600625=2425.
    Thus, sin2θ=2sinθcosθ=27252425=336625 Since sin(BAC)=BCAC, then BC=ACsin(BAC)=625sin2θ=625336625 and so BC=336.

    Answer: 336

  5. Since the circle has diameter 219, it has radius 19.
    Let AP=2x. Thus, BQ=2AP=4x.
    Let CD=2y and let M be the midpoint of CD. Thus, CM=MD=y.
    Join A to R. Since AB is a diameter, then ARB=90.
    Since quadrilateral PQRA has right angles at P, Q and R, then it must have four right angles, and so it is a rectangle.
    Thus, AR=PQ=PC+CD+DQ=1+2y+1=2y+2 and RQ=AP=2x.
    Since BQ=4x, then RB=BQRQ=4x2x=2x.
    By the Pythagorean Theorem in ARB, we obtain AR2+RB2=AB2(2y+2)2+(2x)2=(219)2(y+1)2+x2=19 Join O to M.

    Since M is the midpoint of chord CD, then OM is perpendicular to CD.
    Since AR is parallel to CD (opposite sides in rectangle PQRA), then OM is also perpendicular to AR, crossing AR at T.
    Consider ATO and ARB. These triangles are similar because they share a common angle at A and each is right-angled (at T and R, respectively).
    Since AOAB=12 (radius and diameter), then TORB=12 and so TO=12RB=x.

    Therefore, OM=MT+TO=2x+x=3x. This is because MT=PA since MT is parallel to side PA of rectangle PQRA and has its endpoints on opposite sides of the rectangle.
    By the Pythagorean Theorem in OMC, we have CM2+OM2=CO2y2+(3x)2=(19)2(since CO is a radius)y2+9x2=19 Since (y+1)2+x2=19, then multiplying by 9 gives 9(y+1)2+9x2=171.
    Subtracting y2+9x2=19 gives 9(y2+2y+1)y2=171198y2+18y143=0(4y13)(2y+11)=0 Since y>0, then y=134. Therefore, x2=19(y+1)2=19(174)2=3041628916=1516.
    Since x>0, then x=154. Finally, AP=2x=152.

    Answer: 152

  6. We use R to represent a red marble, B to represent blue, and G to represent green.
    Since there are 15 marbles of which 3 are red, 5 are blue, and 7 are green, there are 15!3!5!7! orders in which the 15 marbles can be removed.
    Since red is the first colour to have 0 remaining, the last of the 15 marbles removed must be B or G. We look at these two cases.

    Suppose that the final marble is B.
    The remaining 4 B’s are thus placed among the first 14 marbles. There are (144) ways in which this can be done.
    This leaves 10 open spaces. The last of these 10 marbles cannot be R, otherwise the last G would be chosen before the last R. Thus, the last of these 10 marbles is G.
    The remaining 6 G’s are thus placed among the remaining 9 spaces. There are (96) ways in which this can be done.
    The 3 R’s are then placed in the remaining 3 open spaces.
    Therefore, there are (144)(96) orders when the final marble is B.

    Suppose that the final marble is G.
    The remaining 6 G’s are thus placed among the first 14 marbles. There are (146) ways in which this can be done.
    This leaves 8 open spaces. The last of these 8 marbles cannot be R, otherwise the last B would be chosen before the last R. Thus, the last of these 8 marbles is B.
    The remaining 4 B’s are thus placed among the remaining 7 spaces. There are (74) ways in which this can be done.
    The 3 R’s are then placed in the remaining 3 open spaces.
    Therefore, there are (146)(74) orders when the final marble is G.

    Thus, the total number of ways in which the first colour with 0 remaining is red equals (144)(96)+(146)(74).

    This means that the desired probability, p, is p=(144)(96)+(146)(74)15!3!5!7!=14!4!10!9!6!3!+14!6!8!7!4!3!15!3!5!7!=14!15!9!10!5!4!7!6!3!3!+14!15!7!8!7!6!5!4!3!3!=115110571+11518751=730+724=28120+35120=63120=2140

    Answer: 2140

PART B

    1. Points A and B are the x-intercepts of the parabola.
      To find their coordinates, we set y=0 to obtain the equation 0=x2+16, which gives x2=16 and so x=±4.
      Therefore, A has coordinates (4,0) and B has coordinates (4,0).

    2. We start by finding the coordinates of M and N.
      Since these points lie on the horizontal line with equation y=7 and on the parabola with equation y=x2+16, we equate expressions for y to obtain 7=x2+16 which gives x2=9 and so x=±3.
      Thus, M has coordinates (3,7) and N has coordinates (3,7).
      Trapezoid MNBA has parallel, horizontal sides MN and AB.
      Here, MN=3(3)=6 and AB=4(4)=8.
      Also, the height of MNBA is 7 since MN lies along y=7 and AB lies along y=0.
      Therefore, the area of MNBA is 12(6+8)7=49.

    3. The origin, O, has coordinates (0,0).
      The vertex, V, of the parabola has coordinates (0,16), since it lies on the y-axis and substituting x=0 into the equation y=x2+16 gives y=16.
      When we set y=33 in the equation of the parabola y=x2+16, we obtain x2=49 and so x=±7.
      The area of quadrilateral VPOQ can be found by adding the areas of VOP and VOQ.
      Each of these triangles can be thought of as having a vertical base VO of length 16, and a horizontal height.
      The length of each horizontal height is 7, since the distance from each of P and Q to the y-axis is 7.
      Therefore, the area of VPOQ is 212167=112.

    1. We begin by noting that a2+a=23 is equivalent to a2+a=49.
      Multiplying this equation by 9 and re-arranging, we obtain the equivalent equation 9a2+9a4=0.
      Factoring, we obtain (3a1)(3a+4)=0 and so a=13 or a=43.
      Since a>0, then a=13.
      Substituting into the original equation, we obtain a2+a=(13)2+13=19+13=49=23 and so a=13 is the only solution to the equation when a>0.

    2. Expanding and simplifying, (m+12)2+(m+12)=m2+m+14+m+12=m2+2m+34 Since m is a positive integer, then m2+2m+34>m2.

      Also, m2+2m+34<m2+2m+1=(m+1)2.

      Thus, m2<m2+2m+34<(m+1)2 which means that the closest perfect square is either m2 or (m+1)2.
      The difference between m2+2m+34 and m2 is 2m+34.
      The difference between m2+2m+34 and (m+1)2 is 14.
      Since m is a positive integer, 2m+34>14, and so (m+12)2+(m+12) is closest to (m+1)2 and their difference is 14.

    3. A fact that will be important throughout this solution is that the expresssion c+c increases as c increases.
      In other words, when 0<c<d, then c+c<d+d.
      This is true since when 0<c<d, then c<d and so c+c<d+d.

      Before proving a general result, we determine the number of positive integers c that satisfy the inequality for each of n=1, n=2, and n=3.

      When n=1, the inequality is 1<c+c<2.
      Since each part is positive, we can square each part and obtain the equivalent inequality 1<c+c<4.
      We see that 1+1=2 and 3<2+2<4 (because 2 is between 1 and 2), and 3+3>4 (because 3>1).
      Thus, when n=1, there are 2 values of c that work.

      When n=2, the inequality is 2<c+c<3 which is equivalent to 4<c+c<9.
      We note that 2+2<4 (because 2<2), that 4<3+3<5 (because 3 is between 1 and 2), that 8<6+6<9 (because 6 is between 2 and 3), and that 9<7+7 (because 7>2).
      Since c+c is increasing, the inequality 4<c+c<9 is true for exactly the positive integers c=3,4,5,6, since it is true for c=3 and c=6 and so is true for all integers in between.

      When n=3, the inequality is 3<c+c<4 which is equivalent to 9<c+c<16.
      Using similar reasoning, we obtain that this inequality is true for the positive integers c=7,8,9,10,11,12.

      Therefore, for n=1,2,3, we obtain 2, 4 and 6 solutions.
      Based on these values of n, we guess that for a general value of n, there are 2n values of c that work. Also, for each n the largest value of c that works appears to be c=n(n+1) and the smallest value of c that works appears to be c=n(n1)+1. (Check to see that these conjectures match the specific cases above.)

      From earlier, we know that the inequality n<c+c<n+1 is equivalent to the inequality n2<c+c<n2+2n+1.
      Our strategy now is to show that the inequality n2<c+c<n2+2n+1 (I) is false when c=n(n1), (II) is true when c=n(n1)+1, (III) is true when c=n(n+1), and (IV) is false when c=n(n+1)+1.
      Since the expression c+c is increasing, these facts will show that the positive integers c for which the inequality is true are exactly the values of c with n(n1)+1cn(n+1), of which there are n(n+1)(n(n1)+1)+1=n2+n(n2n+1)+1=2n and so the number of c will be even, as required.

      Step (I):

      We note that when n1, we have n(n1)=n2n<n2 and so n(n1)<n2=n.
      Thus, when c=n(n1), we see that c+c=n(n1)+n(n1)=n2n+n(n1)<n2n+n=n2 and so the inequality n2<c+c<n2+2n+1 is false.

      Step (II):

      Next, we note that when n1, we have n(n1)+1=n2n+1>n22n+1 and so n(n1)+1>n22n+1=n1.
      Also, when n1, we have n(n1)+1=n2n+1n2 and so n(n1)+1n2=n.
      Thus, when c=n(n1)+1, we see that c+c=n(n1)+1+n(n1)+1=n2n+1+n(n1)+1>n2n+1+(n1)=n2 Also, when c=n(n1)+1, we see that c+c=n(n1)+1+n(n1)+1n2n+1+n=n2+1<n2+2n+1 Thus, the inequality n2<c+c<n2+2n+1 is true.

      Connecting (I) and (II) to (III) and (IV):

      We have shown that, for all integers n1, n(n1)+n(n1)<n2 and n2<(n(n1)+1)+n(n1)+1<n2+2n+1 This means that, for all integers m1, m(m1)+m(m1)<m2 and m2<(m(m1)+1)+m(m1)+1<(m+1)2 Setting m=n+1 and restricting n1 (and so m2), we obtain n(n+1)+n(n+1)<(n+1)2() and (n+1)2<(n(n+1)+1)+n(n+1)+1<(n+2)2 This last set of inequalities shows that n2<c+c<(n+1)2 is false when c=n(n+1)+1, which is item (IV).
      Since n(n+1)>n2, we can update () to n2<n(n+1)+n(n+1)<(n+1)2 which is item (III).

      We have now shown the desired inequalities for c=n(n1), c=n(n1)+1, c=n(n+1), and c=n(n+1)+1, which means that the number of positive integers c that satisfy the inequality for a given n is 2n, which is even, as required.

    1. We note that if (a,b,c,d)=(A,B,C,D) satisfies the equation ab=cd (that is, if AB=CD), then so do (a,b,c,d)=(B,A,D,C) and (a,b,c,d)=(C,D,A,B) and (a,b,c,d)=(D,C,B,A) since BA=DC,CD=AB,DC=BA Let g(n) be the number of quadruples (a,b,c,d) of distinct integers from Sn for which ab=cd with a<b and c<d and a<c.
      Then f(n)=4g(n) since each of the quadruples counted by g(n) can be transformed into four quadruples counted by f(n) by using the four arrangements above.
      We note that every quadruple counted by f(n) has either a<b and c<d or a>b and c>d (because ab=cd and so the signs of ab and cd are the same) and has either a<c or a>c (because ac).
      Therefore, the quadruples counted by f(n) can be grouped into sets of four quadruples exactly one of which satisfies a<b and c<d and a<c.

      To determine the value of f(6), we first determine the value of g(6).
      To determine the value of g(6), we look at the possible pairs (a,b) with a<b and for each of these count the number of possible pairs (c,d).
      We work through pairs (a,b) by increasing value of ba, recalling that c<d and a<c:

      • (a,b)=(1,2): the possibilities for (c,d) are (3,4), (4,5), (5,6)

      • (a,b)=(2,3): the possibilities for (c,d) are (4,5), (5,6)

      • (a,b)=(3,4): the only possibility for (c,d) is (5,6)

      • (a,b)=(1,3): the possibilities for (c,d) are (2,4), (4,6)

      • (a,b)=(2,4): the only possibility for (c,d) is (3,5)

      • (a,b)=(3,5): the only possibility for (c,d) is (4,6)

      • (a,b)=(1,4): the possibilities for (c,d) are (2,5), (3,6)

      • (a,b)=(2,5): the only possibility for (c,d) is (3,6)

      • (a,b)=(1,5): the only possibility for (c,d) is (2,6)

      Since there are thus 3+2+1+2+1+1+2+1+1=14 quadruples (a,b,c,d), we have g(6)=14 and so f(6)=56.

    2. To answer parts (b) and (c), we determine an explicit expression for g(n) (and hence for f(n)) in terms of n. With this information in hand, we can then answer the given questions.
      Suppose that n is an even positive integer. We write n=2m for some positive integer m1.
      To determine an expression for g(2m), we count the number of quadruples (a,b,c,d) of distinct integers from Sn for which ab=cd with a<b and c<d and a<c.
      Since a<b and a<c and bc, then either b<c or b>c.
      This gives us two cases for the ordering of a,b,c,d: first that a<b<c<d and second that a<c<b<d. Note that d>b in both cases because the “gap” between a and b is equal to the gap between c and d.

      Case 1: a<b<c<d

      Suppose that ba=dc=t for some positive integer t.
      Since da=dc+ca>dc+ba=2t and da2m1, then 2t2m1, which means that tm12. Since t and m are integers, then tm1.
      We count the number of quadruples (a,b,c,d) with a<b<c<d for each value of t from 1 to m1, inclusive.
      Suppose that t=1.
      Here, b=a+1 and the pair (c,d) can be as small as (a+2,a+3) and as large as (2m1,2m).
      For the pair (a,b)=(a,a+1), there are 2m(a+3)+1=2ma2 pairs (c,d).
      Since a can range from a=1 to a=2m3 (the latter of which gives the quadruple (2m3,2m2,2m1,2m)), there is a total of (2m3)+(2m4)++2+1=12(2m3)(2m2) quadruples (a,b,c,d) in the case t=1.

      More generally, suppose that 1tm1.
      Here, b=a+t and the pair (c,d) can be as small as (a+t+1,a+2t+1) and as large as (2mt,2m).
      For the pair (a,b)=(a,a+t), there are thus 2m(a+2t+1)+1=2ma2t pairs (c,d).
      Since a can range from a=1 to a=2m2t1 (the latter of which gives the quadruple (2m2t1,2mt1,2mt,2m)), there is a total of (2m2t1)+(2m2t2)++2+1=12(2m2t1)(2m2t) quadruples (a,b,c,d) for each t with 1tm1.

      We note that 12(2m2t1)(2m2t)=(2m2t1)(mt)=2m22mt2mt+2t2m+t=2t2+t(4m+1)+(2m2m) From the Useful Fact, when we add 2t2 for each t from 1 to m1, we obtain 2(m1)m(2m1)6.
      Adding t(4m+1) for each t from 1 to m1 is equivalent to multiplying (4m+1) by the sum of the integers from 1 to m1, which gives (4m+1)12(m1)m.
      Adding (2m2m) for each t from 1 to m1 gives (m1)(2m2m).
      Adding these partial sums together, we obtain the total number of triples in this case, which is 2(m1)m(2m1)6+(4m+1)12(m1)m+(m1)(2m2m)=m(m1)(2m13+4m+12+(2m1))=m(m1)(4m26+12m+36+12m66))=m(m1)4m56

      Case 2: a<c<b<d

      Suppose that ba=dc=t for some positive integer t.
      Since 2mdb+1, then t=ba2m11=2m2. Since a<c<b, then ba+2, which means that t2.
      Thus, the range of values for t is 2t2m1.
      For each pair of values (a,b), the value of c can, in theory, move between a+1 and b1 with the restriction that d=c+t must still be at most 2m.

      Our strategy is to work through the possible values of t, in each case starting with a=1 to establish the possible quadruples in this case when a=1, and then count the number of ways these relative positions for (a,b,c,d) can be slid until d reaches 2m.

      Suppose that t=2 and a=1. Here, b=a+t=3. Since a<c<b, then c=2 which gives d=4.
      The base quadruple (a,b,c,d)=(1,3,2,4) can have each component increased by 1 until we obtain the largest possible quadruple with this relative positioning, which is (a,b,c,d)=(2m3,2m1,2m2,2m).
      There are 2m4+1=2m3 quadruples in this case.

      Suppose that t=3 and a=1. Here, b=4 and so c=2 or c=3.
      This gives the base quadruples (a,b,c,d)=(1,4,2,5) and (a,b,c,d)=(1,4,3,6).
      There are 2m4 and 2m5 possible quadruples of these two forms.

      Next, consider a general value of t with 2tm, along with a=1.
      Here, b=t+1 and so 2ct.
      This gives the base quadruples (a,b,c,d)=(1,t+1,c,c+t) for 2cm.
      For each of these values of c, the corresponding value of d satisfies d2m, since d=c+tm+m=2m.
      For a given value of c, there are 2m(c+t)+1=2mct+1 such quadruples.
      As c increases from 2 to t, we see that there are (2mt1)+(2mt2)++(2m2t+2)+(2m2t+1) quadruples. There are t1 terms in this sum.
      This sum can be re-written as (2mt1)+(2mt2)++(2m2t+2)+(2m2t+1)=((2m2t)+(t1))+((2m2t)+(t2))++((2m2t)+2)+((2m2t)+1)=(t1)(2m2t)+(1+2++(t2)+(t1))=(t1)(2m2t)+12t(t1)=2mt2t22m+2t+12t212t=32t2+t(2m+32)2m When t=1, this expression is equal to 0.
      Thus, we can add these expressions for t=1 to t=m and obtain the same sum as adding from t=2 to t=m.
      This sum is thus 32(12+22++m2)+(1+2++m)(2m+32)2mm=32m(m+1)(2m+1)6+4m+32m(m+1)22m2=m(m+1)(2m+1)4+m(m+1)(4m+3)42m2=m(m+1)(2m+2)42m2=m2((m+1)24m)=m2(m22m+1)=m(m1)22

      Finally, consider a general value of t with m+1t2m2.
      Again, (a,b,c,d)=(1,t+1,c,c+t) and 2ct, but c+t2m means that c2mt2(t1)t=t2 which is strictly less than the upper bound coming from the value of b.
      In other words, when t is large enough, not all of the values of c between 2 and t actually produce admissible values of d.
      As above, there are 2mct+1 triples that work for a given value of c, but the upper bound for c is different.
      As c increases from 2 to 2mt, we see that there are (2mt1)+(2mt2)++2+1 quadruples, which equals 12(2mt1)(2mt).
      We need to add these expressions for t=m+1 to t=2m2.
      We substitute s=2m1t, which means that we need to add 12s(s+1)=12s2+12s for s=1 to s=m2.
      This gives 12(12+22+(m2)2)+12(1+2++(m2))=12(m2)(m1)(2m3)6+12(m2)(m1)2=(m2)(m1)4(2m33+1)=(m2)(m1)42m3=(m2)(m1)m6 This means that g(2m)=m(m1)4m56+m(m1)22+(m2)(m1)m6=m(m1)6((4m5)+3(m1)+m2)=m(m1)6(8m10)=m(m1)(4m5)3 and so f(2m)=4g(2m)=4m(m1)(4m5)3.
      Using this formula,

      • when m=1, we obtain f(2)=0,

      • when m=2, we obtain f(4)=42133=8, and

      • when m=3, we obtain f(6)=43273=56,

      which agrees with the specific results that we found earlier.

      Thus, f(40)=42019753=38000, which is the answer to (b).
      (Note that we could have found this value by calculating the value of g(40) specifically and directly.)

    3. Lastly, we find two even positive integers n=2m with n<2022 (that is, with m<1011) for which 2022 is a divisor of f(n)=f(2m).
      To do this, we note first that 2022=21011=23337 and 337 is a prime number. (How could you verify that 337 is prime?)
      Therefore, we need to find two positive integers m<1011 for which 4m(m1)(4m5)3 is a multiple of 2, of 3, and of 337.
      We can assume that m3, and so we can write m as one of m=3q or m=3q+1 or m=3q+2 for some positive integer q.
      When m=3q, we have 4m(m1)(4m5)3=4(3q)(3q1)(12q5)3=4q(3q1)(12q5) When m=3q+1, we have 4m(m1)(4m5)3=4(3q+1)(3q)(12q1)3=4q(3q+1)(12q1) When m=3q+2, we have 4m(m1)(4m5)3=4(3q+2)(3q+1)(12q+3)3=4(3q+1)(3q+2)(4q+1) Each of these expressions is a positive integer since it is the product of positive integers.
      Each of these expressions is even, since each has a factor of 4 in it.
      Therefore, we need to determine 2 small enough values of q for which one of these expressions are divisible by 3 and by 337.
      Since 337 is a larger prime, we work by looking for factors of 337 first and then checking for factors of 3.

      The first two positive multiples of 337 are 337 and 674.
      We can check by working through the factors of these three expressions that it is impossible for one of the factors to equal 337 and for another factor in the same expression to be a multiple of 3.
      When q=224, we obtain 3q+2=674 (which is a multiple of 337) and 4q+1=897 (which is a multiple of 3).
      Therefore, when q=224, the expression 4(3q+1)(3q+2)(4q+1) is a multiple of 2022, and so setting m=3224+2=674 tells us that f(1348) is a multiple of 2022.
      When q=225, we obtain 3q1=674 (which is a multiple of 337) and q=225 (which is a multiple of 3).
      Therefore, when q=225, the expression 4q(3q1)(12q5) is a multiple of 2022, and so setting m=3225=675 tells us that f(1350) is a multiple of 2022.
      This gives us the answer to (c): two even integers n with n<2022 for which f(n) is a multiple of 2022 are n=1348 and n=1350.