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2016 Canadian Team Mathematics Contest
Solutions

May 2016

© 2016 University of Waterloo

Individual Problems

  1. Evaluating, 20+200+2010+20160=1+1+1+1=4.

    Answer: 4

  2. Since 20 minutes is 13 of an hour and Zeljko travelled at 30 km/h for 13 of an hour, then Zeljko travelled 3013=10 km during this portion of the trip.
    Since 30 minutes is 12 of an hour and Zeljko travelled at 20 km/h for 12 of an hour, then Zeljko travelled 2012=10 km during this portion of the trip.
    In total, Zeljko travelled 10+10=20 km.

    Answer: 20

  3. Using the definition, 2(25)=2(2552)=2(3225)=27=2772=12849=79

    Answer: 79

  4. We call the numbers tossed on the two dice A and B.
    There are 6 possible values for each of A and B, and each such value is equally likely.
    Therefore, there are 66=36 equally likely possible pairs of outcomes.
    If A=1, then for A+B<10, B can equal any of 1,2,3,4,5,6.
    If A=2, then for A+B<10, B can equal any of 1,2,3,4,5,6.
    If A=3, then for A+B<10, B can equal any of 1,2,3,4,5,6.
    If A=4, then for A+B<10, B can equal any of 1,2,3,4,5.
    If A=5, then for A+B<10, B can equal any of 1,2,3,4.
    If A=6, then for A+B<10, B can equal any of 1,2,3.
    Thus, 6+6+6+5+4+3=30 pairs of outcomes satisfy the requirement and so the probability that A+B is less than 10 is 3036 or 56.

    Answer: 56

  5. An integer is divisible by 15 exactly when it is divisble by both 5 and 3.
    For a positive integer to be divisible by 5, its units (ones) digit must be 5 or 0.
    For a positive integer to be a palindrome, its first and last digits must be the same.
    Since a positive integer cannot have a first digit of 0, then the first and last digits of a palindrome that is divsible by 5 must be 5.
    Therefore, a five-digit palindrome that is divisible by 15 (and hence by 5) must be of the form 5aba5. (Note that the thousands and tens digits must be the same.)
    For 5aba5 to be as large as possible, we start with a=b=9.
    This gives 59995 which is not divisible by 15. (To test this, we can use a calculator or use a test for divisibility by 3, since 59995 is already divisible by 5.)
    The next largest five-digit palindrome of the form 5aba5 is 59895, which is divisible by 15. (If we reduced the value of a rather than the value of b, the resulting palindrome would be smaller than this one.)
    Therefore, the largest five-digit palindrome that is divisible by 15 is 59895.

    Answer: 59895

  6. Since this arithmetic sequence of integers includes both even and odd integers, then the common difference in the sequence must be odd. (If it were even, then all of the terms in the sequence would be even or all would be odd.)
    Suppose that this common difference is d.
    Note that 555 is one of the terms in the sequence.
    If d=1, then either or both of 554 and 556 would be in the sequence. Since no term other than 555 can contain repeated digits, then d1.
    If d=3, then either or both of 552 and 558 would be in the sequence. Since no term other than 555 can contain repeated digits, then d3.
    If d=5, then 550 cannot be in the sequence, so 555 would have to be the smallest term in the sequence. This would mean that 560 and 565 were both in the sequence. Since no term other than 555 can contain repeated digits, then d5.
    We can also rule out the possibility that d9:

    Since there are 14 terms in the sequence and the common difference is d, then the largest term is 13d larger than the smallest term.
    Since all terms are between 500 and 599, inclusive, then the maximum possible difference between two terms is 99.
    This means that 13d99. Since d is an odd integer, then cannot be 9 or greater.

    Since d is odd, d cannot be 1, 3 or 5, and d7, then it must be the case that d=7.
    Can we actually construct a sequence with the desired properties?
    Starting with 555 and adding and subtracting 7s, we get the sequence 506,513,520,527,534,541,548,555,562,569,576,583,590,597 This sequence has 14 terms in the desired range, including 7 even and 7 odd terms, and no term other than 555 has repeated digits.
    (Note that this sequence cannot be extended in either direction and stay in the desired range.)
    Therefore, the smallest of the 14 house numbers must be 506.

    Answer: 506

  7. Since Q(a+1,4a+1) lies on the line with equation y=ax+3, then 4a+1=a(a+1)+30=a23a+20=(a1)(a2) Since a>1, then we must have a=2.
    Therefore, Q has coordinates (3,9).
    We want to find the coordinates of points P and R so that, together with O(0,0) and Q(3,9), the quadrilateral OPQR is a square.
    Since OQ is a diagonal of the square, then the midpoint, M, of OQ is the centre of the square.

    M is at the intersection of OQ and PR, and angle OMR equals 90 degrees.

    Since O has coordinates (0,0) and Q has coordinates (3,9), then M has coordinates (32,92).
    Note that PM and MR are perpendicular to OM, since the diagonals of a square are perpendicular.
    Since the slope of OM is 920320=3, then the slope of PM and MR is 13.
    The equation of the line through M with slope 13 is y92=13(x32) or y=13x+5.
    An arbitrary point on this line can be written in the form A(3t,t+5) for some real number t.
    The points P and R will be the two points on this line for which OP and PQ are perpendicular and OR and RQ are perpendicular.
    Thus, given an arbitrary point A on the line, we want to find all t for which OA and AQ are perpendicular (that is, for which the slopes of OA and AQ have a product of 1).
    The slope of OA is t+53t. The slope of AQ is (t+5)93t3.
    We have the following equivalent equations: t+53tt43t3=1(t+5)(t4)=3t(3t3)t2t20=9t2+9t10t210t20=0t2t2=0(t2)(t+1)=0 Thus, t=1 or t=2.
    Substituting into (3t,t+5), we obtain P(3,6) and R(6,3).
    We can check that the points O(0,0), P(3,6), Q(3,9), R(6,3) give OP=PQ=QR=RO and that consecutive sides are perpendicular, so OQPR is a square.
    Thus, the coordinates of the points P and R are (3,6) and (6,3).

    Answer: (3,6),(6,3)

  8. Suppose that Claudine has N candies in total.
    When Claudine divides all of her candies equally among 7 friends, she has 4 candies left over. This means that N is 4 more than a multiple of 7, or N=7a+4 for some non-negative integer a.
    When Claudine divides all of her candies equally among 11 friends, she has 1 candy left over. This means that N is 1 more than a multiple of 11, or N=11b+1 for some non-negative integer b.
    Note then that N+10=7a+14=7(a+2) and N+10=11b+11=11(b+1).
    In other words, N+10 is a multiple of 7 and a multiple of 11.
    This means that N+10 is a multiple of 77, or N is 10 less than a multiple of 77.
    Since Claude has p packages containing 19 candies each, then Claudine has 19p candies.
    Putting these pieces together, we want to find the smallest non-negative integer p for which 19p is 10 less than a multiple of 77.
    To do this efficiently, we list, in increasing order, the positive integers that are 10 less than a multiple of 77 until we reach one that is a multiple of 19: 67,144,221,298,375,452,529,606,683,760 Since 760 is the first number in this list that is a multiple of 19, then 760 is the smallest multiple of 19 that is 4 more than a multiple of 7 and 1 more than a multiple of 11.
    This means that the smallest possible value of p is 76019 or 40.

    Answer: 40

  9. By symmetry, the area of the shaded region on each of the four trapezoidal faces will be the same. Therefore, the total shaded area will be 4 times the shaded area on trapezoid EFBA.
    Let h be the height of trapezoid EFBA. In other words, h is the distance between the parallel sides EF and AB.
    Let P be the point of intersection of EB and FA.

    Since EFGH is a square with side length 1, then EF=1. Similarly, AB=7.
    Therefore, the area of trapezoid EFBA is 12(1+7)h=4h.
    The shaded area on trapezoid EFBA equals the area of EFBA minus the area of APB.
    Since EF and AB are parallel, then FEP=PBA.
    This means that FPE is similar to APB.
    Since AB:EF=7:1, then the ratio of the height of APB to the height of FPE must also be 7:1.
    Since the sum of the heights of these triangles is the height of the trapezoid, then the height of APB is 78h.
    Thus, the area of APB is 12AB78h=4916h, because AB=7.
    Therefore, the shaded area on trapezoid EFBA is 4h4916h=1516h, and so the total shaded area on the truncated pyramid is 41516h=154h.
    To complete our solution, we need to calculate h.
    To do this, we begin by dropping perpendiculars from E and F to points Q and R on AB, from F to point S on BC, and from F to point T in square ABCD.

    Since EF is parallel to QR and EQR=FRQ=90, then EFRQ is a rectangle.
    Thus, QR=1.
    Since EQ=FR=h and EA=FB and each of EQA and FRB is right-angled, then these triangles are congruent, and so AQ=RB.
    Since AB=7 and QR=1, then AQ=RB=3. Similarly, BS=3.
    Now RBST must be a square of side length 3, since EF lies directly above ST extended and FG lies directly above RT extended.
    Finally, we have TR=3, FT=4, FR=h, and FTR=90.
    By the Pythagorean Theorem, h2=32+42=25, and so h=5, since h>0.
    This means that the total shaded area is 154h=754.

    Answer: 754

  10. The inequality x4+8x3y+16x2y2+168x2+32xy is equivalent to x4+16x2y2+16+8x3y8x232xy0 We note that (x2+4xy4)2=(x2)2+(4xy)2+(4)2+2x2(4xy)+2x2(4)+2(4xy)(4)=x4+16x2y2+16+8x3y8x232xy Therefore, the original inequality is equivalent to (x2+4xy4)20.
    Since the square of any real number is at least 0, then this inequality is equivalent to (x2+4xy4)2=0 or x2+4xy4=0.
    Similarly, since (y28xy+5)2=y4+64x2y2+25+10y216xy380xy, then the inequality y4+64x2y2+10y2+2516xy3+80xy is equivalent to (y28xy+5)20, which is equivalent to y28xy+5=0.
    Thus, the original system of inequalities is equivalent to the system of equations x2+4xy4=0y28xy+5=0 Adding 5 times the first of these equations to 4 times the second equation we obtain 5x2+20xy20+4y232xy+20=0 or 5x212xy+4y2=0.
    Factoring, we obtain (5x2y)(x2y)=0, which gives x=25y or x=2y.
    If x=25y, then x2+4xy4=0 becomes 425y2+85y24=0 or 425y2+4025y2=4 which gives 4425y2=4 or y2=10044=2511, and so y=±511.
    Since x=25y, then x=±211.
    In this case, we get the pairs (x,y)=(211,511),(211,511).
    If x=2y, then x2+4xy4=0 becomes 4y2+8y24=0 which gives 12y2=4 or y2=13, and so y=±13.
    Since x=2y, then x=±23.
    In this case, we get the pairs (x,y)=(23,13),(23,13).
    Therefore, the pairs that satisfy the original system of inequalities are (x,y)=(211,511),(211,511),(23,13),(23,13)

    Answer: (211,511),(211,511),(23,13),(23,13)

Team Problems

  1. Evaluating, 2+01×6=2+06=4.

    Answer: 4

  2. Since the average of 5 numbers is 7, then the sum of the 5 numbers is 5×7=35.
    Therefore, 3+5+6+8+x=35 or 22+x=35 and so x=13.

    Answer: 13

  3. Since 1.6=1, then 2.3+1.6=2.3+1=1.3=2.

    Answer: 2

  4. Starting with ABC, the magician

    The net effect of these 3 moves is to swap the second and third cups.
    When the magician goes through this sequence of moves 9 times, the net effect starting from ABC is that he swaps the second and third cups 9 times.
    Since 9 is odd, then the final order is ACB.

    Answer: ACB

  5. Since (1,2) lies on the parabola with equation y=ax2+bx+c, then the coordinates of the point satisfy the equation of the parabola.
    Thus, 2=a(12)+b(1)+c or a+b+c=2.

    Answer: 2

  6. Since 211=2048 and 28=256, then 21128=2048256=1792.
    Therefore, m=11 and n=8, which gives m2+n2=112+82=121+64=185.
    Alternatively, we can factor the left side of the equation 2m2n=1792 to obtain 2n(2mn1)=1792.
    Since m>n, then 2mn1 is an integer.
    Now, 1792=287.
    Since 2n(2mn1)=287, then 2n=28 (which gives n=8) and 2mn1=7 (which gives mn=3 and so m=11).

    Answer: 185

  7. The number of two-digit integers in the range 10 to 99 inclusive is 9910+1=90.
    We count the number of two-digit positive integers whose tens digit is a multiple of the units (ones) digit.
    If the units digit is 0, there are no possible tens digits.
    If the units digit is 1, the tens digit can be any digit from 1 to 9. This gives 9 such numbers.
    If the units digit is 2, the tens digit can be 2,4,6,8. This gives 4 such numbers.
    If the units digit is 3, the tens digit can be 3,6,9. This gives 3 such numbers.
    If the units digit is 4, the tens digit can be 4,8. This gives 2 such numbers.
    If the units digit is any of 5,6,7,8,9, the only possibility is that the tens digit equals the units digit. This gives 5 such numbers.
    In total, there are 9+4+3+2+5=23 two-digit positive integers whose tens digit is a multiple of the units digit, and so the probability that a randomly chosen two-digit integer has this property is 2390.

    Answer: 2390

  8. The area of pentagon ADCBZ equals the area of rectangle ABCD minus the area of AZB.
    Since the area of rectangle ABCD is 2016, then ABCB=2016.
    The area of AZB equals 12(AB)(ZH).
    Since ZH:CB=4:7, then ZH=47CB.
    Thus, the area of AZB is 12(AB)(47CB)=27(AB)(CB)=27(2016)=576.
    Therefore, the area of pentagon ADCBZ is 2016576=1440.

    Answer: 1440

  9. The integer with digits AAA is equal to 100A+10A+A=111A.
    Similarly, the integer with digits AAB is equal to 110A+B, the integer with digits ABB is equal to 100A+11B, and the integer with digits BBB is equal to 111B.
    From the given addition, we obtain 111A+(110A+B)+(100A+11B)+111B=1503 or 321A+123B=1503.
    Dividing both sides by 3, we obtain 107A+41B=501.
    Rearranging gives 41B=501107A.
    Since A and B are positive integers, then 107A<501.
    Since 1075=535, then the possible values of A are 1,2,3,4.
    If A=1, then 501107A=394 which is not a multiple of 41.
    If A=2, then 501107A=287 which equals 7×41.
    (Neither A=3 nor A=4 makes 501107A a multiple of 41.)
    Therefore, A=2 and B=7.
    (Checking, we see that 222+227+277+777=1503.)
    Thus, A3+B2=23+72=8+49=57.

    Answer: 57

  10. Suppose that, on the way from Appsley to Bancroft, Clara rides x km downhill, y km on level road, and z km uphill.
    Then on the reverse trip, she travels x km uphill, y km on level road, and z km downhill.
    Clara rides downhill at 24 km/h, on level road at 16 km/h, and uphill at 12 km/h.
    Since the original trip takes 2 hours, then x24+y16+z12=2.
    Since the reverse trip takes 2 hours and 15 minutes (or 2.25 hours), then x12+y16+z24=2.25.
    Adding these two equations, we obtain (124+112)x+(116+116)y+(112+124)z=4.25 Since 124+112=124+224=324=18, then the last equation is equivalent to x8+y8+z8=4.25 Multiplying both sides by 8, we obtain x+y+z=34.
    The distance from Appsley to Bancroft is (x+y+z) km, or 34 km.

    Answer: 34

  11. We write out the first few terms of the sequence using the given rule: 4,5,5+14=32,3/2+15=12,1/2+13/2=1,1+11/2=4,4+11=5,5+14=32 Since the 6th and 7th terms are equal to the 1st and 2nd terms, respectively, then the sequence repeats with period 5. (This is because two consecutive terms determine the next term, so identical pairs of consecutive terms give identical next terms.)
    Since 1234=246(5)+4, then the 1234th term is the 4th term after a cycle of 5 and so equals the 4th term, which is 12.

    Answer: 12

  12. Since Austin chooses 2 to start and Joshua chooses 3 next, we can list all of the possible orders of choices of numbers:

    Order of Numbers Joshua’s 1st Rd Score Austin’s 1st Rd Score Joshua’s 2nd Rd Score Austin’s 2nd Rd Score Joshua’s Total Austin’s Total
    2, 3, 1, 4, 5 6 3 4 20 10 23
    2, 3, 1, 5, 4 6 3 5 20 11 23
    2, 3, 4, 1, 5 6 12 4 5 10 17
    2, 3, 4, 5, 1 6 12 20 5 26 17
    2, 3, 5, 1, 4 6 15 5 4 11 19
    2, 3, 5, 4, 1 6 15 20 4 26 19

    Since Austin’s total is larger than Joshua’s total in 4 of the 6 games, then the probability that Austin wins is 46 or 23.

    Answer: 23

  13. Suppose that the cone has radius r, height h, and slant height s.

    r, s, and h form a triangle where r is the base, h is the height, and s is the hypotenuse.

    Since the lateral surface area of the cone is 80π and the total surface area is 144π, then the area of the base of the cone is 144π80π=64π.
    Thus, πr2=64π, which gives r2=64 or r=8 (since r>0).
    Since the lateral surface area is 80π and r=8, then πrs=80π or 8s=80 which gives s=10.
    Now, by the Pythagorean Theorem, s2=r2+h2 and so h2=s2r2=10282=36.
    Since h>0, then h=6.
    Thus, the volume of the cone is 13πr2h=13π82(6)=128π.
    Suppose that a sphere of equal volume has radius R. Then 43πR3=128π.
    Thus, R3=34128=96, and so R=963.

    Answer: 963

  14. Using logarithm rules, log3(1115)+log3(1114)+log3(1113)++log3(118)+log3(117)+log3(116)=log3(1415)+log3(1314)+log3(1213)+log3(1112)+log3(1011)+log3(910)+log3(89)+log3(78)+log3(67)+log3(56)=log3(1415131412131112101191089786756)=log3(515)=log3(13)=1

    Answer: 1

  15. We want to determine the number of four-digit positive integers of the form 4xyz that are divisible by 45.
    A positive integer is divisible by 45 exactly when it is divisible by both 5 and 9.
    For an integer to be divisible by 5, its units digit must be 0 or 5.
    Thus, such numbers are of the form 4xy0 or 4xy5.
    For an integer to be divisible by 9, the sum of its digits must be divisible by 9.
    Case 1: 4xy0
    The sum of the digits of 4xy0 is 4+x+y.
    Since x and y are between 0 and 9, inclusive, then 0x+y18.
    For 4+x+y to be divisible by 9, we need x+y=5 or x+y=14.
    If x+y=5, then (x,y)=(0,5),(1,4),(2,3),(3,2),(4,1),(5,0).
    If x+y=14, then (x,y)=(5,9),(6,8),(7,7),(8,6),(9,5).
    There are 11 possible four-digit numbers in this case.
    Case 1: 4xy5
    The sum of the digits of 4xy5 is 9+x+y.
    Since x and y are between 0 and 9, inclusive, then 0x+y18.
    For 9+x+y to be divisible by 9, we need x+y=0 or x+y=9 or x+y=18.
    If x+y=0, then (x,y)=(0,0).
    If x+y=9, then (x,y)=(0,9),(1,8),(2,7),(3,6),(4,5) and their reverses.
    If x+y=18, then (x,y)=(9,9).
    There are 12 possible four-digit numbers in this case.
    Overall, there are 11+12=23 possible combinations.

    Answer: 23

  16. Since A1A2A3An is a regular n-gon, then A1A2A3=A2A3A4=A3A4A5=180360n.
    (This comes from the fact that the sum of the interior angles in a regular n-gon is 180(n2).)
    Therefore, PA2A3=PA4A3=180(180360n)=360n.
    Also, the reflex angle at A3 equals 360(180360n)=180+360n.
    Since the sum of the angles in quadrilateral PA2A3A4 equals 360, then 120+2(360n)+180+360n=360 or 1080n=60 and so n=18.

    Answer: 18

  17. There are (213)=21(20)(19)3!=7(10)(19) ways of choosing 3 of the 21 marbles from the bag.
    We count the number of ways in which 3 marbles of the same colour can be chosen.
    There are 0 ways of choosing 3 magenta or 3 puce marbles, since there are fewer than 3 marbles of each of these colours.
    There is 1 way of choosing 3 cyan marbles since there are exactly 3 cyan marbles in total.
    Since there are 4 ecru marbles, there are (43)=4 ways of choosing 3 ecru marbles.
    Since there are 5 aquamarine marbles, there are (53)=5(4)(3)3(2)(1)=10 ways of choosing 3 aquamarine marbles.
    Since there are 6 lavender marbles, there are (63)=6(5)(4)3(2)(1)=20 ways of choosing 3 lavender marbles.
    Thus, there are 1+4+10+20=35 ways of choosing 3 marbles of the same colour.
    Since there are 7(10)(19) ways of choosing 3 marbles, then the probability that all three are the same colour is 357(10)(19)=7(5)7(10)(19)=12(19)=138.
    For this probability to equal 1k, we must have k=38.

    Answer: 38

  18. We begin by comparing pairs of the given expressions:

    Therefore,

    Noting that 5<274<223, we combine these relationships to get that

    This tells us that f(x)=3x2 when x5, and f(x)=2x+3 when 5x223, and f(x)=25x when x223.
    Since f(x) has positive slope when x5, then the maximum value of f(x) when x5 occurs when x=5. We have f(5)=3(5)2=13.
    Since f(x) has positive slope when 5x223, then the maximum value of f(x) when 5x223 occurs when x=223. We have f(223)=2223+3=533.
    Since f(x) has negative slope when x223, then the maximum value of f(x) when x223 occurs when x=223. We already know that f(223)=533.
    Comparing the maximum values of f(x) over the three intervals, we can conclude that the maximum value of f(x) is 533, since 533>13.

    Answer: 533

  19. Suppose that DBC=θ. Then ABC=2θ.

    Now AD=ACDC=AC1.
    Also, tan(ABC)=tan2θ=ACBC, so AC=BCtan2θ.
    In DBC, by the Pythagorean Theorem, we have BC=BD2DC2=3212=8=22 Therefore, AD=22tan2θ1.
    But tanθ=tan(DBC)=DCBC=122.
    This means that tan2θ=2tanθ1tan2θ=1/21(1/8)=1/27/8=872.
    Finally, we obtain AD=22tan2θ1=162721=1671=97.

    Answer: 97

  20. Solution 1
    Let T be the total number of animals.
    Suppose that, in the first line, there are C cows and H horses.
    Suppose that, in the second line, there are w cows and x horses that are standing opposite the cows in the first line, and that there are y cows and z horses that are standing opposite the horses in the first line.
    Since there are 75 horses, then H+x+z=75.
    Since there are 10 more cows opposite cows than horses opposite horses, then w=z+10.
    The total number of animals is T=C+H+w+x+y+z.
    But y+z=H and C=w+x so T=(w+x)+H+w+x+H=2(H+w+x).
    Now w=z+10 so T=2(H+z+10+x)=20+2(H+x+z)=20+2(75)=170, and so the total number of animals is 170.

    Solution 2
    Let h be the number of pairs consisting of a horse opposite a horse.
    Let c be the number of pairs consisting of a cow opposite a cow.
    Let x be the number of pairs consisting of a horse opposite a cow.
    Since the total number of horses is 75, then 75=2h+x. (Each “horse-horse” pair includes two horses.)
    Also, the given information tells us that c=h+10.
    The total number of animals is thus 2h+2c+2x=2h+2(h+10)+2(752h)=2h+2h+20+1504h=170

    Answer: 170

  21. Suppose that the three small circles have centres A, B and C.
    For the larger circle to be as small as possible, these three small circles must just touch the larger circle (which has centre O) at P, Q and S, respectively.
    Note that AP=BQ=CS=1.

    Let r be the radius of the larger circle. Then OP=OQ=OS=r.
    Since the larger circle is tangent to each of the smaller circles at P, Q and S, this means that the circles share a common tangent at each of these points.
    Consider the point P. Since AP is perpendicular to the tangent to the smaller circle at P and OP is perpendicular to the tangent to the larger circle at P and this tangent is common, then AP and OP must be parallel. Since AP and OP both pass through P, then AP lies along OP.
    Since AP=1 and OP=r, then OA=r1.
    In a similar way, we can find that OB=OC=r1.
    Now since the circles with centres A and B just touch, then the distance AB equals the sum of the radii of the circles, or AB=2.
    Similarly, AC=BC=2.
    Consider AOB. We have OA=OB=r1 and AB=2.
    By symmetry, AOB=BOC=COA=13(360)=120.
    Let M be the midpoint of AB. Thus, AM=MB=1.

    Triangle AOB.

    Since AOB is isosceles, then OM is perpendicular to AB and bisects AOB.
    This means that AOM is a 30-60-90 triangle.
    Therefore, AOAM=23 and so r11=23 or r=1+23.
    Thus, the radius of the smallest circle enclosing the other three circles is 1+23

    Answer: 1+23

  22. If we impose no restrictions initially, each of the 7 patients can be assigned to one of 3 doctors, which means that there are 37 ways to assign the patients to doctors.
    To determine the number of ways in which the patients can be assigned to doctors so that each doctor is assigned at least one patient, we subtract from 37 the number of ways in which at least one doctor has 0 patients.
    This can happen if all of the patients are assigned to one doctor. There are 3 ways to do this (all patients assigned to Huey, all patients assigned to Dewey, or all patients assigned to Louie).
    This can also happen if all of the patients are assigned to two doctors and each of these two doctors are assigned at least one patient.
    If each of the 7 patients are assigned to Huey and Dewey, for example, then there are 2 choices for each patient, giving 27 ways minus the 2 ways in which all of the patients are assigned to Huey and in which all of the patients are assgined to Dewey. This gives 272 ways of assigning the patients in this way.
    Similarly, there are 272 ways of assigning the patients to Huey and Louie, and 272 ways of assigning the patients to Dewey and Louise.
    Overall, this means that there are 3733(272)=218733(126)=1806 ways of assigning the patients to the doctors.

    Answer: 1806

  23. Suppose that S=1+3a+5a2+7a3+.
    Then aS=a+3+5a+7a2+9a3+.
    Therefore aSS=a+2+2a+2a2+2a3+.
    Since a>1, then 2+2a+2a2+2a3+ is an infinite geometric series with common ratio 1a<1.
    Therefore, 2+2a+2a2+2a3+=211a=2aa1.
    Substituting, we obtain aSS=a+2aa1 and so (a1)S=(a2a)+2aa1 which gives (a1)S=a2+aa1 or S=a2+a(a1)2.

    Answer: a2+a(a1)2

  24. Let P0 be the point on the line through AM that minimizes the distance from C.
    Then CP0 is perpendicular to AM.
    (Note that any other point P on this line would form CP0P with right-angle at P0, making CP the hypotenuse of the triangle and so the longest side. In particular, any other point P on the line gives CP>CP0.)
    Consider AMC.

    The area of AMC equals 12AC2c=ACc. This is because M lies directly above AC, which is a diagonal of the base of the prism, and so the height of AMC equals the height of the prism, which is 2c.
    Also, the area of AMC equals 12AMh, where h is the perpendicular distance from C to AM. (Here, we are thinking of AM as a base of the triangle.)
    But CP0 is the corresponding height, so h=CP0.
    In other words, 12AMCP0=ACc, and so CP0=2ACcAM.
    So we need to determine the length of AC and the length of AM.
    AC is the hypotenuse of right-angled ABC.
    Since AB=2a and BC=AD=2b, then AC=AB2+BC2=(2a)2+(2b)2=4a2+4b2=2a2+b2 AM is the hypotenuse of right-angled AFM.
    Since AF=2c and FM=12FH=12AC=a2+b2, then AM=AF2+FM2=4c2+a2+b2 Therefore, CP0=2(2a2+b2)c4c2+a2+b2=4ca2+b2a2+b2+4c2

    Answer: 4ca2+b2a2+b2+4c2

  25. If m is a positive integer that ends in 9999, then m+1 must end in 0000.
    This means that m+1 is a multiple of 10 000; that is, m+1=10000k=104k for some positive integer k.
    Since m does not end in five or more 9s, then m+1 does not end in 00000 and so the units digit of k is not 0. In other words, k is not divisible by 10.
    We determine the first several terms in each sequence to see if we can find a pattern.
    By definition, t1=1 and t2=m.
    Next, s2=t1+t2=m+1=104k.
    This gives t3=3s2=3104k.
    Next, s3=t1+t2+t3=s2+t3=104k+3104k=4104k.
    This gives t4=4s3=44104k.
    Next, s4=s3+t4=4104k+4(4104k)=54104k.
    This gives t5=5s4=554104k.
    Next, s5=s4+t5=54104k+5(54104k)=654104k.
    This gives t6=6654104k.
    Based on this apparent pattern, we guess that, for n5, we have tn=nn(n1)54104k=nn!104k3! and sn1=n(n1)54104k=n!10k3!.
    We will proceed based on this guess, which we will prove at the end of this solution.
    Following this guess, t30=3030!104k3!=530!104k.
    We want to find all positive integers N for which 530!104k=N! for some positive integer k that is not divisible by 10.
    We note that 30! must be a divisor of N! and so N30.
    We also know that N! contains at least five more factors of 5 than 30! since N!=30!55424k.
    This means that N must be enough larger than 30 so that N! includes at least five more factors of 5.
    We note that 35!=35(34)(33)(32)(31)30!, which contains one more factor of 5 than 30! and that 35! is the smallest factorial with this property.
    Similarly, 40! contains one more factor of 5 than 35!.
    Also, 45! contains one more factor of 5 than 40! and is the smallest factorial that includes three more factors of 5 than 30!.
    Since 52 is a divisor of 50, then 50! contains two more factors of 5 than 45! and is the smallest factorial that thus includes five more factors of 5 than 30!. We conclude that N50.
    We note that 50!, 51!, 52!, 53!, and 54! each contain the same number of factors of 5, and that 55! (and thus every N! with N55) contains at least six more factors of 5 than 30!.
    Now since N50, then 50! is a divisor of N!=30!5524k.
    Rearranging, we get N(N1)(N2)(33)(32)(31)=5524k or N(N1)(N2)(33)(2)(31)=55k.
    Since the left side is even, then k is even.
    Since k is even and k is not divisible by 10, then k is not divisible by 5. (That is, the right side contains exactly five factors of 5.)
    This means that N cannot be 55 or greater, as in this case the number of factors of 5 (at least six on the left side) would not balance.
    Therefore, the possible values of N are 50,51,52,53,54.
    To complete our solution, we prove that tn=nn!104k3! and sn1=n!104k3! when n5.
    We already know that t5=554104k=55!104k3! and that s4=54104k=5!104k3!.
    We prove that the desired formulas are true for all n5 by using a technique called mathematical induction.
    We have proven that these formulas are correct for n=5.
    If we can prove that whenever the forms are correct for n=j, then they must be correct for n=j+1, then their correctness for n=5 will give their correctness for n=6 which will give their correctness for n=7 and so on, giving their correctness for all n5.
    So suppose that tj=jj!104k3! and that sj1=j!104k3!.
    Then sj=sj1+tj=j!104k3!+jj!104k3!=j!104k3!(1+j)=(j+1)!104k3!, as expected.
    This means that tj+1=(j+1)sj=(j+1)(j+1)!104k3!=(j+1)(j+1)!104k3!, as expected.
    Therefore, if tj and sj1 have the correct forms, then tj+1 and sj have the correct forms.
    This proves that tn and sn1 have the correct forms for all n5, which completes our solution.

    Answer: 50,51,52,53,54

Relay Problems

(Note: Where possible, the solutions to parts (b) and (c) of each Relay are written as if the value of t is not initially known, and then t is substituted at the end.)

    1. Evaluating, 102×3=106=4.

    2. The area of a triangle with base of length 2t and height of length 3t+1 is 12(2t)(3t+1) or t(3t+1).
      Since the answer to (a) is 4, then t=4, and so t(3t+1)=4(13)=52.

    3. Since AB=BC, then BCA=BAC=t.
      Therefore, ABC=180BCABAC=1802t.
      Since the answer to (b) is 52, then t=52, and so ABC=180252=180104=76

    Answer: 4,52,76

    1. Since x:6=15:10, then x6=1510 which gives x=61510=9.

    2. If 3(x+5)4=t+33x2, then 3(x+5)=4t+2(33x) or 3x+15=4t+66x, which gives 9x=4t9 or x=49t1.
      Since the answer to (a) is 9, then t=9 and so x=49t1=41=3.

    3. We start with the given equation and complete the square: y=3x2+6mx+36=3(x2+2mx+12)=3((x+m)2m+12)=3(x+m)2+(363m) Therefore, the coordinates of the vertex of this parabola are (m,363m).
      Since the y-coordinate of the vertex is t, then 363m=t or 3m=36t and so m=1213t.
      Since the answer to (b) is 3, then t=3, and so m=1213t=121=11.

    Answer: 9,3,11

    1. To find the x-intercept of the line with equation 20x+16y40=0, we set y=0 and get 20x40=0 or x=2.
      To find the y-intercept of the line with equation 20x+16y64=0, we set x=0 and get 16y64=0 or y=4.
      The sum of the intercepts is 2+4=6.

    2. Since BC:CD=2:1, then BC=2CD.
      Since BD=9t, then BC+CD=9t or CD+2CD=9t and so 3CD=9t or CD=3t, which gives BC=6t.

      The area of ACE, which is denoted k, equals the area of trapezoid ABDE minus the areas of ABC and CDE.
      The area of trapezoid ABDE is 12(AB+DE)(BD) or 12(2t+9t)(9t) which equals 992t2.
      The area of ABC is 12(AB)(BC) or 12(2t)(6t) which equals 6t2.
      The area of CDE is 12(CD)(DE) or 12(3t)(9t) which equals 272t2.
      Therefore, k=992t26t2272t2=30t2.
      Since the answer to (a) is 6, then t=6, and so 136k=3036t2=30.

    3. The volume of a cylinder with radius 2 and height a is π(2)2a=2πa.
      The volume of a cylinder with radius 5 and height b is π(5)2b=5πb.
      From the given information, 2πa+5πb=10πt or 2a+5b=10t.
      Since a>0 and b>0, then 2a<10t or a<5t. Also, 5b<10t or b<2t.
      Rearranging 2a+5b=10t gives 2a=10t5b or a=5t52b.
      If we assume that t is a positive integer, then for a to be a positive integer, it must be the case that b is even.
      Since b<2t, then the possible even values of b less than 2t are 2,4,6,,2t4,2t2. There are t1 such values.
      Under the assumption that t is a positive integer, these are exactly the values of b that give positive integer values for a.
      In particular, we obtain the pairs (a,b)=(5t5,2),(5t10,4),,(10,2t4),(5,2t2), of which there are t1.
      Since the answer to (b) is 30, then t=30 (which is a positive integer), and so t1=29. That is, there are 29 pairs of positive integers (a,b) that satisfy the requirements.

    Answer: 6,30,29

    1. Since 93=729 and 103=1000, then the largest positive integer a with a3<999 is a=9.
      Since 25=32 and 35=243, then the smallest positive integer b with b5>99 is b=3.
      Therefore, ab=6.

    2. Suppose that Oscar saw N birds in total.
      From the given information, he saw 35N sparrows and 14N finches.
      Therefore, he saw N35N14N=N1220N520N=320N cardinals.
      Note that Oscar saw 35N sparrows and 320N cardinals. Since 3/53/20=4, then he saw four times as many sparrows as cardinals.
      Since he saw 10t cardinals, then he saw 4×10t=40t sparrows.
      Since the answer to (a) is 6, then t=6, and so Oscar saw 40t=240 sparrows.

    3. Suppose that there are f seats in the first row.
      Since every row after the first contains 4 more seats than the previous, then the remaining 19 rows contain f+4,f+8,f+12,,f+72,f+76 seats.
      The numbers of seats in the 20 rows form an arithmetic sequence with 20 terms, first term equal to f, and last term equal to f+76.
      The total number of seats is equal to the sum of this series, which equals 202(f+(f+76)) or 10(2f+76).
      Since we are told that there are 10t seats in total, then 10t=10(2f+76) or t=2f+76.
      Therefore, 2f=t76 or f=12t38.
      Since the answer to (b) is 240, then f=12t38=12038=82.
      In other words, there are 82 seats in the first row.

    Answer: 6,240,82