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2021 Canadian Senior
Mathematics Contest
Solutions

Wednesday, November 17, 2021
(in North America and South America)

Thursday, November 18, 2021
(outside of North American and South America)

©2021 University of Waterloo


PART A

  1. If the row of 11 squares is built from left to right, the first square requires 4 toothpicks and each subsequent square requires 3 additional toothpicks. These 3 toothpicks form the top edge, the right edge and the bottom edge. The left edge is formed by the right edge of the previous square.
    Therefore, 11 squares require 4+103=34 toothpicks.

    Answer: 34

  2. Using the definition of , 5x=30(5+1)(x2)=306(x2)=30x2=5 and so x=7.

    Answer: x=7

  3. The slope of PR is 2010=2.

    Therefore, the line that is parallel to PR has the form y=2x+b.

    The midpoint of QR has coordinates (4+12,0+22) which is the point (52,1).
    Since the line with equation y=2x+b passes through the point with coordinates (52,1), then 1=252+b, which gives 1=5+b and so b=4.

    Answer: 4

  4. Throughout this solution, we convert the statements about probability to fractions of the populations in question.
    Since 2000 people are asked about side effects and 325 of these people have severe side effects, then the number of people with severe side effects is 3252000=240.
    Since 240 have severe side effects and 23 of them were given Medicine A, then the number of people who were given Medicine A that have severe side effects is 23240=160.
    This means that the number of people who were given Medicine B and have severe side effects is 240160=80.

    Medicine A Medicine B TOTAL
    Mild side effects
    Severe side effects 160 80 240
    TOTAL

    Since 1000 people were given Medicine A and 19100 have severe or mild side effects, then the number of people who were given Medicine A that have severe or mild side effects is 190.
    This means that the number of people who were given Medicine A that have mild side effects is 190160=30.

    Medicine A Medicine B TOTAL
    Mild side effects 30
    Severe side effects 160 80 240
    TOTAL 190

    Similarly, the number of people who were given Medicine B that have severe or mild side effects is 3201000=150, of whom 80 have severe side effects and so 15080=70 have mild side effects.

    Medicine A Medicine B TOTAL
    Mild side effects 30 70
    Severe side effects 160 80 240
    TOTAL 190 150

    Since 30 people with mild side effects were given Medicine A and 70 were given Medicine B, then the probability that a random person with mild side effects was given Medicine B is 7070+30=70100=710.

    Answer: 710

  5. Solution 1

    Since loga(bc)=clogab, then log2(x2)=2log2xlog2(x3)=3log2x2logx8=2logx(23)=23logx2=6logx220logx(32)=20logx(25)=205logx2=100logx2 Let t=log2x.
    Since x is also the base of a logarithm in the equation, then x1, which means that t0.
    Therefore, logx2=log2logx=1(logx)/(log2)=1log2x=1t Starting with the original equation, we obtain the following equivalent equations: log2(x2)+2logx8=392log2(x3)+20logx(32)2log2x+6logx2=3923log2x+100logx22t+6t=3923t+100t(using t=log2x)2t2+6t=392t3t2+100(since t0)t2+3t=196t3t2+100(t2+3)(3t2+100)=196t23t4+109t2+300=196t23t487t2+300=0t429t2+100=0(t225)(t24)=0(t+5)(t5)(t+2)(t2)=0 Therefore, t=5 or t=5 or t=2 or t=2.
    Thus, log2x=5 or log2x=5 or log2x=2 or log2x=2.
    Thus, x=25=132 or x=25=32 or x=22=14 or x=22=4.

    We can check by substitution that each of these values of x is indeed a solution to the original equation.

    Solution 2

    Let a=log2x and b=logx2.
    Then 2a=x and xb=2, which gives 2ab=(2a)b=xb=2.
    Since 2ab=2, then ab=1.
    As in Solution 1, log2(x2)=2log2x=2alog2(x3)=3log2x=3a2logx8=6logx2=6b20logx(32)=100logx2=100b Starting with the original equation, we thus obtain the following equivalent equations: log2(x2)+2logx8=392log2(x3)+20logx(32)2a+6b=3923a+100b(2a+6b)(3a+100b)=392(a+3b)(3a+100b)=1963a2+109ab+300b2=1963a2+60ab+300b2=19649ab3a2+60ab+300b2=147(since ab=1)a2+20ab+100b2=49(a+10b)2=72 and so a+10b=7 or a+10b=7.
    Since ab=1, then b=1a and so a+10a=7 or a+10a=7.
    These equations give a2+10=7a (or equivalently a27a+10=0) and a2+10=7a (or equivalently a2+7a+10=0).
    Factoring, we obtain (a2)(a5)=0 or (a+2)(a+5)=0 and so a=2,5,2,5.

    Since a=log2x, then x=25=132 or x=25=32 or x=22=14 or x=22=4.
    We can check by substitution that each of these values of x is indeed a solution to the original equation.

    Answer: x=132,32,14,4

  6. Let the side length of the square base ABCD be 2a and the height of the pyramid (that is, the distance of P above the base) be 2h.
    Let F be the point of intersection of the diagonals AC and BD of the base. By symmetry, P is directly above F; that is, PF is perpendicular to the plane of square ABCD.
    Note that AB=BC=CD=DA=2a and PF=2h. We want to determine the value of 2a.
    Let G be the midpoint of FC.
    Join P to F and M to G.

    Consider PCF and MCG.
    Since M is the midpoint of PC, then MC=12PC.

    Since G is the midpoint of FC, then GC=12FC.

    Since PCF and MCG share an angle at C and the two pairs of corresponding sides adjacent to this angle are in the same ratio, then PCF is similar to MCG.
    Since PF is perpendicular to FC, then MG is perpendicular to GC.
    Also, MG=12PF=h since the side lengths of MCG are half those of PCF.

    The volume of the square-based pyramid PABCD equals 13(AB2)(PF)=13(2a)2(2h)=83a2h.

    Triangular-based pyramid MBCD can be viewed as having right-angled BCD as its base and MG as its height.

    Thus, its volume equals 13(12BCCD)(MG)=16(2a)2h=23a2h.

    Therefore, the volume of solid PABMD, in terms of a and h, equals 83a2h23a2h=2a2h.
    Since the volume of PABMD is 288, then 2a2h=288 or a2h=144.
    We have not yet used the information that BMD=90.
    Since BMD=90, then BMD is right-angled at M and so BD2=BM2+MD2.
    By symmetry, BM=MD and so BD2=2BM2.
    Since BCD is right-angled at C, then BD2=BC2+CD2=2(2a)2=8a2.
    Since BGM is right-angled at G, then BM2=BG2+MG2=BG2+h2.
    Since BFG is right-angled at F (the diagonals of square ABCD are equal and perpendicular), then BG2=BF2+FG2=(12BD)2+(14AC)2=14BD2+116AC2=14BD2+116BD2=516BD2=52a2 Since 2BM2=BD2, then 2(BG2+h2)=8a2 which gives 52a2+h2=4a2 or h2=32a2 or a2=23h2.

    Since a2h=144, then 23h2h=144 or h3=216 which gives h=6.
    From a2h=144, we obtain 6a2=144 or a2=24.
    Since a>0, then a=26 and so AB=2a=46.

    Answer: 46

PART B

    1. The expression x24 is a difference of squares which can be factored as x24=(x+2)(x2)

    2. Using (a) with x=98, we obtain 9824=(98+2)(982)=10096 and so k=96.
      Alternatively, 9824=96044=9600=10096 which gives k=96.

    3. If (20n)(20+n)=391, then 202n2=391.
      This gives 400n2=391 from which we obtain n2=9 and so n=±3.
      Since n is positive, then n=3.
      We can verify that 1723=391.

    4. We note that 3999991=40000009=2000232=(2000+3)(20003)=20031997.
      Since we have written 3999991 as the product of two positive integers that are each greater than 1, we can conclude that 3999991 is not a prime number.
       

    1. Consider a Leistra sequence with a1=216.
      In the sequence, a2 must be an even integer of the form a2=a1d=216d where d is an integer between 10 and 50, inclusive.
      Since 216=63=23×33, the positive divisors of 216 are 1,2,3,4,6,8,9,12,18,24,27,36,54,72,108,216 From this list, the divisors of 216 that are between 10 and 50 are 12, 18, 24, 27, 36.
      The quotients when 216 is divided by these integers are 18, 12, 9, 8, 6, respectively.
      Since a2 is even, then we can have a2=18 (with d=12) or a2=12 (with d=18) or a2=8 (with d=27) or a2=6 (with d=36).
      If a2=18, there is no possible a3 since the only divisor of a2=18 that is greater than 10 is 18 and this would give a3=1 which is not even.
      If a2=12, there is no possible a3 since the only divisor of a2=12 that is greater than 10 is 12 and this would give a3=1 which is not even.
      If a2=8 or a2=6, there is no divisor larger than 10.
      This means that no Leistra sequence starting with a1=216 can have more than two terms.
      Therefore, there are four Leistra sequences with a1=216, namely 216,18 and 216,12 and 216,8 and 216,6.

    2. Consider a Leistra sequence with a1=2×350.
      In the sequence, a2 must be an even integer of the form a2=a1d1=2×350d1 where d1 is an integer between 10 and 50, inclusive.
      Since a1 includes only one factor of 2 and since a2 must be even, d1 cannot itself be even.
      Therefore, d1 must be an odd divisor of a1=2×350 that is between 10 and 50, inclusive.
      The odd positive divisors of a1=2×350 are the integers of the form 3j with 0j50; in other words, the odd positive divisors of a1 are powers of 3.
      The first few powers of 3 are 3, 9, 27, 81. The only such power between 10 and 50 is 33=27. Therefore, the only possibility for d1 is d1=33 which gives a2=a1d1=2×347.
      The next term in the sequence is an even positive integer a3 of the form a3=a2d2 where d2 is an integer between 10 and 50, inclusive. (While we can find divisors between 10 and 50 to divide out and still obtain an even integer, (P3) tells us that the sequence must continue.)
      Using a similar argument, d2=33 and so a3=2×344.
      Since each subsequent term ai is found by dividing its previous term by an integer, then each ai is itself a divisor of a1 which means the divisors of each ai are a subset of the divisors of the original a1.
      In this case, this means that the only divisor that we will ever be able to use in this sequence is 33 and we must divide this out until we can no longer do so.
      This means that the Leistra sequence continues in the following way: 2×350,2×347,2×344,,2×38,2×35 where the last term listed is the 16th term so far.
      We can again divide out a factor of 33 to obtain 2×350,2×347,2×344,,2×38,2×35,2×32 The last term is now 18; as in (a), the sequence cannot be continued.
      Therefore, with a1=2×350, the sequence is completely determined without choice, and so there is exactly 1 Leistra sequence with a1=2×350.

    3. Consider Leistra sequences with a1=22×350.
      The divisors that can be used to construct the Leistra sequence are divisors of 22×350 that leave an even quotient.
      Since a1 includes 2 factors of 2, these divisors can thus include 0 or 1 factor of 2 in order to preserve the parity of the quotient.
      This means that the divisors are of the form 3j or 2×3j and are between 10 and 50, inclusive.
      From (b), the only possible divisor of the form 3j in this range is 33=27.
      Also, the integers of the form 2×3j with j=1,2,3 are 6, 18, 54, and so the only possible divisor in the desired range is 18.
      This means that this Leistra sequence can only use the divisors 27 and 18. We note further that the divisor 18 can be used at most once, otherwise the remaining quotient (and thus next term) would be odd.
      Suppose that the divisor 18 is not used.
      Using a similar argument to (b), this gives the sequence 22×350,22×347,22×344,,22×38,22×35,22×32 The last term in this list is 22×32=36. This term cannot be the final term in the sequence because 3618=2, and so if 36 were the final term, (P3) would be violated.

      Thus, the sequence cannot use 27 as its only divisor and must use 18 at least once and at most once (thus exactly once).
      Therefore, any Leistra sequence beginning with a1=22×350 must use the divisor 18 exactly once and the divisor 27 a number of times.
      If the sequence above is extended by using the divisor 18 to generate a final term, we obtain 22×350,22×347,22×344,,22×38,22×35,22×32,2 In this case, the sequence uses the divisor 33=27 a total of 16 times and the divisor 18 once.
      The divisor 33=27 cannot be used 17 times, since (33)17=351 and we cannot divide 3 out of 2×350 a total of 51 times and still obtain an integer quotient.
      Also, since (33)15×(2×32)=2×347 and 22×3502×347=2×33, the divisor 33=27 has to be used more than 15 times, since if it used only 15 times, the sequence must be continued by dividing again by 27.
      Therefore, the Leistra sequences starting with a1=22×350 all have 18 terms, and are generated by dividing 16 times by 27 and 1 time by 18. Any order of these divisors will generate a sequence of even integers, and so will generate a Leistra sequence.
      There are 17 ways of arranging these 17 divisors as there are 17 places to choose to place the 18 and the remaining places are filled with the 27s.
      Since there are 17 ways of arranging the divisors, there are 17 Leistra sequences with a1=22×350.

    4. Consider Leistra sequences with a1=23×350.
      The divisors that can be used to construct the Leistra sequence are divisors of 23×350 that leave an even quotient.
      Since a1 includes 3 factors of 2, these divisors can thus include 0, 1 or 2 factors of 2 in order to preserve the parity of the quotient.
      From (c), these divisors are 18 and 27 along with any divisors of the form 22×3j that are between 10 and 50, inclusive.
      Since the integers of the form 22×3j with j=0,1,2,3 are 4, 12, 36, 108, the possible divisors in the desired range are 12 and 36.
      Thus, these Leistra sequences are generated using the divisors 27, 18, 12, and 36. To count the number of Leistra sequences, we count the number of ways of arranging the possible divisors, noting that at most 2 factors of 2 can be included amongst the set of divisors used and at most 50 factors of 3 can be included amongst the set of divisors used.
      Looking at the number of factors of 2, we can see that either (i) the divisors are all odd, (ii) one divisor of 12 is used, (iii) one divisor of 36 is used, (iv) two divisors of 18 are used, or (v) one divisor of 18 is used. Any other combination of the even divisors will not satisfy the conditions of Leistra sequences.

      Case 1: 27 only
      Note that 2716=(33)16=348 which is a divisor of a1.
      Since 27=33, we cannot use 17 or more 27s, as a1 includes only 50 factors of 3.
      Also, we cannot use 15 or fewer 27s, as the sequence can be continued by dividing by another 27.
      Since a1(33)16=23×350348=23×32, then the sequence cannot use just 27s since we could continue the sequence that did by dividing by 12, for example.
      Therefore, there are no sequences in this case.

      Case 2: 27 and 12
      The divisor 12 can only be used once, otherwise too many factors of 2 would be removed.
      Since 12×2715=22×346 and 12×2716=22×349 and 12×2717=22×352, then the divisor of 27 must be used exactly 16 times. (If 27 is used 17 times, too many 3s are removed; if 27 is used 15 times, the sequence can be extended by dividing by 27 again. If 27 is used 16 times, the quotient is 23×35022×349=2×3=6 which ends the sequence.)
      Therefore, Leistra sequences can be formed by using exactly 16 divisors equal to 27 and 1 divisor equal to 12.
      Since these divisors can be used in any order, there are 17 such sequences, as in (c).

      Case 3: 27 and 36
      The divisor 36 can only be used once, otherwise too many factors of 2 would be removed.
      Since 36×2715=22×347 and 36×2716=22×350 and 36×2717=22×353, then the divisor of 27 must be used exactly 16 times.
      Therefore, Leistra sequences can be formed by using exactly 16 divisors equal to 27 and 1 divisor equal to 36.
      Since these divisors can be used in any order, there are 17 such sequences, as in (c).

      Case 4: 27 and two 18s
      Note that 18 cannot be used more than two times and that 18 cannot be combined with 12 or 36, otherwise the remaining quotient would be either odd or not an integer.
      Since 182×2714=22×346 and 182×2715=22×349 and 182×2716=22×352, then the divisor of 27 must be used exactly 15 times.
      Therefore, Leistra sequences can be formed by using exactly 15 divisors equal to 27 and 2 divisors equal to 18.
      There are (172)=17×162=136 ways of choosing 2 of the 17 positions in the sequence of divisors for the 18s to be placed. The remaining spots are filled with 27s.
      Therefore, there are 136 such sequences.

      Case 5: 27 and one 18
      Since 18×2715=2×347 and 18×2716=2×350 and 18×2717=2×353, then the divisor of 27 must be used exactly 16 times, leaving a final term of 22=4.
      As in Case 2 and Case 3, there are 17 such sequences.

      Having considered all possibilities, there are 17+17+136+17=187 Leistra sequences with a1=23×350.

    1. Using f(x+y)=f(x)g(y)+g(x)f(y) with x=y=0, we obtain f(0)=2f(0)g(0).
      From this, we get f(0)(2g(0)1)=0.
      Thus, f(0)=0 or g(0)=12.

      Using g(x+y)=g(x)g(y)f(x)f(y) with x=y=0, we obtain g(0)=(g(0))2(f(0))2.

      If g(0)=12, we obtain 12=14(f(0))2 which gives (f(0))2=14.

      Since f(0) is real, then (f(0))20 and so (f(0))214.
      Thus, g(0)12, which means that f(0)=0.

      Since f(a)0 for some real number a, setting x=a and y=0 gives f(a+0)=f(a)g(0)+g(a)f(0) Since f(0)=0, we obtain f(a)=f(a)g(0).
      Since f(a)0, we can divide by f(a) to obtain g(0)=1.
      Therefore, f(0)=0 and g(0)=1.
      (We note that the functions f(x)=sinx and g(x)=cosx satisfy the given conditions, so there does exist at least one Payneful pair of functions.)

    2. Solution 1

      Let t be a real number.
      From (a), f(0)=0 and g(0)=1. Thus, (f(0))2+(g(0))2=1 and so 1=(f(t+(t)))2+(g(t+(t)))2=(f(t)g(t)+g(t)f(t))2+(g(t)g(t)f(t)f(t))2=(f(t))2(g(t))2+2f(t)g(t)g(t)f(t)+(g(t))2(f(t))2   +(g(t))2(g(t))22g(t)g(t)f(t)f(t)+(f(t))2(f(t))2=(f(t))2(g(t))2+2f(t)f(t)g(t)g(t)+(g(t))2(f(t))2   +(g(t))2(g(t))22f(t)f(t)g(t)g(t)+(f(t))2(f(t))2=(f(t))2(g(t))2+(g(t))2(g(t))2+(f(t))2(f(t))2+(g(t))2(f(t))2=(g(t))2((f(t))2+(g(t))2)+(f(t))2((f(t))2+(g(t))2)=((f(t))2+(g(t))2)((f(t))2+(g(t))2)=h(t)h(t) Since h(t)h(t)=1 for every real number t, setting t=5 gives h(5)h(5)=1, as required.

      Solution 2

      Let t be a real number. Then h(t)h(t)=((f(t))2+(g(t))2)((f(t))2+(g(t))2)=(f(t))2(f(t))2+(f(t))2(g(t))2+(g(t))2(f(t))2+(g(t))2(g(t))2 Since f(0)=0, then f(t+(t))=0 which gives 0=f(t)g(t)+g(t)f(t) and so f(t)g(t)=g(t)f(t).
      Since g(0)=1, then g(t+(t))=1 which gives g(t)g(t)f(t)f(t)=1.
      Squaring both sides, we obtain 1=(g(t)g(t)f(t)f(t))2=(g(t))2(g(t))22g(t)g(t)f(t)f(t)+(f(t))2(f(t))2=(f(t))2(f(t))2f(t)g(t)g(t)f(t)f(t)g(t)g(t)f(t)+(g(t))2(g(t))2=(f(t))2(f(t))2f(t)g(t)(f(t)g(t))(g(t)f(t))g(t)f(t)+(g(t))2(g(t))2     (using f(t)g(t)=g(t)f(t) twice)=(f(t))2(f(t))2+(f(t))2(g(t))2+(g(t))2(f(t))2+(g(t))2(g(t))2=h(t)h(t) from above. Note that this is true for every real number t.
      Setting a=5, we obtain h(5)h(5)=1, as required.

    3. From (b), setting t=2021, we obtain h(2021)h(2021)=1.
      We will show that h(2021)=1.
      Since h(2021)h(2021)=1, then h(2021)0 and h(2021)0.
      Suppose that h(2021)1.
      Since h(x)=(f(x))2+(g(x))2, then h(x)0 for all real numbers x.
      Also, since 10f(x)10 and 10g(x)10 for all real numbers x, then h(x)=(f(x))2+(g(x))2102+102=200 for all real numbers x.
      Since h(2021)1 and h(2021)0, then either h(2021)>1 and 0<h(2021)<1, or h(2021)>1 and 0<h(2021)<1.
      Now, for any real number s, f(2s)=f(s+s)=f(s)g(s)+g(s)f(s)=2f(s)g(s) and g(2s)=g(s+s)=g(s)g(s)f(s)f(s)=(g(s))2(f(s))2 and so h(2s)=(f(2s))2+(g(2s))2=(2f(s)g(s))2+((g(s))2(f(s))2)2=4(f(s))2(g(s))2+(g(s))42(f(s))2(g(s))2+(f(s))4=(f(s))4+2(f(s))2(g(s))2+(g(s))4=((f(s))2+(g(s))2)2=(h(s))2 Since h(2s)=(h(s))2, then h(4s)=(h(2s))2=((h(s))2)2=(h(s))4 and h(8s)=(h(4s))2=((h(s))4)2=(h(s))8 In general, if h(2ks)=(h(s))2k for some positive integer k, then h(2k+1s)=h(22ks)=((h(s))2k)2=(h(s))2k+1 This shows that h(2ns)=(h(s))2n for every positive integer n. (We have used a process here called mathematical induction.)
      Now, we know that h(2021)>1 or h(2021)>1.
      If h(2021)>1, let b=2021 and c=h(2021).
      If h(2021)>1, let b=2021 and c=h(2021).
      Since c>1, then cm grows without bound as m increases.
      In particular, cm>200 when m>logc200.
      From above, h(2nb)=c2n for every positive integer n.
      Therefore, when 2n>logc200, we have h(2nb)>200, which is not possible since we saw above that h(x)200 for all real numbers x.
      This is a contradiction, which means that the assumption that h(2021)1 cannot be true, and so h(2021)=1. (In fact, modifying the argument above shows that h(x)=1 for all real numbers x.)