CEMC Banner

2017 Cayley Contest
Solutions
(Grade 10)

Tuesday, February 28, 2017
(in North America and South America)

Wednesday, March 1, 2017
(outside of North American and South America)

©2016 University of Waterloo


  1. Evaluating, 6×1112×111=666222=444.
    Alternatively, we note that 6×1112×111=(62)×111=111(4)=444.

    Answer: (C)

  2. Evaluating, 52953=2592=162=8.
    Alternatively, we note that 529=5232=(53)(5+3) and so 52953=(53)(5+3)53=5+3=8

    Answer: (D)

  3. The height of the snowman equals the sum of the lengths of the diameters of the three spheres.
    Since the radii of the spheres are 10 cm, 20 cm and 30 cm, then the lengths of their diameters are 20 cm, 40 cm and 60 cm.
    Thus, the height of the snowman is 20 cm+40 cm+60 cm=120 cm.

    Answer: (D)

  4. We write each of the choices in lowest terms: 4444455555=4(11111)5(11111)=4555556666=5(1111)6(1111)=56 666777=6(111)7(111)=677788=7(11)8(11)=7889 (The last choice was already in lowest terms.)
    Next, we note that 45=115 and 56=116 and 67=117 and 78=118 and 89=119.
    The fraction with the greatest value will be the one that is equal to 1 minus the smallest fraction.
    Since 19<18<17<16<15, then the fraction with the greatest value is 89.

    Answer: (E)

  5. Since 300 litres drains in 25 hours, then the rate at which water is leaving the tank equals 300 L25 h or 12 L/h.

    Answer: (A)

  6. When Penelope folds the paper in half, the number of layers doubles.
    Starting with 4 layers of paper, then after the next three folds, there are 8 and then 16 and then 32 layers of paper. Additional folds create more layers.
    Of the given choices (which are all less than 32), only 16 is a possible number of layers.

    Answer: (D)

  7. By definition, 27=22(7)2(72)=4(7)2(49)=2898=70.

    Answer: (B)

  8. In option (A), the first and third cards have 0 numbers in common, so (A) is not correct.
    In option (B), the second and third cards have 2 numbers in common, so (B) is not correct.
    In option (C), the first and third cards have 2 numbers in common, so (C) is not correct.
    In option (E), the first and third cards have 2 numbers in common, so (E) is not correct.
    In option (D), the first and second cards share exactly 1 number (namely, 4), the first and third cards share exactly one number (namely, 7), and the second and third cards share exactly one number (namely, 2). Thus, (D) is correct.

    Answer: (D)

  9. Since the bill including a 13% tip was $226, then $226 is 113% of the bill before tax.
    Thus, the bill before tax was $2261.13=$200 before tax.
    The amount of the tip is 15% of the bill before tax, or $200×0.15=$30.

    Answer: (C)

  10. Since PQR is a straight angle, then PQT+RQT=180.
    Therefore, x+(x50)=180 and so 2x50=180 or 2x=230, which gives x=115.
    Therefore, TUR=(x+25)=140.
    Since TU and PS are parallel, then URS and TUR are alternating angles, which means that URS=TUR=140.

    Answer: (B)

  11. Since 10 identical squares have a total area of 160 cm2, then the area of each square is 160 cm210 or 16 cm2.
    Since the area of each of the 10 identical squares is 16 cm2, then the side length of each of these squares is 16 cm2=4 cm.
    The perimeter of the given figure equals 22 square side lengths (4 on the left side, 4 on the bottom, 4 on the right side, 2 separate ones on the top, and 8 in the “U” shape in the middle).
    Therefore, the perimeter of the figure is 4 cm×22=88 cm.

    Answer: (C)

  12. Since the mean of p, q and r is 9, then p+q+r3=9 and so p+q+r=27.
    Since the mean of s and t is 14, then s+t2=14 or s+t=28.
    Therefore, the mean of the five integers is p+q+r+s+t5=(p+q+r)+(s+t)5=27+285, which equals 11.

    Answer: (A)

  13. First, we consider the column of units (ones) digits.
    From this column, we see that the units digit of Z+Z+Z (or 3Z) must be 5.
    By trying the possible digits from 0 to 9, we find that Z must equal 5.
    When Z=5, we get 3Z=15, and so there is a “carry” of 1 to the column of tens digits.
    From this column, we see that the units digit of 1+Y+Y+Y (or 3Y+1) must be 7.
    By trying the possible digits from 0 to 9, we find that Y must equal 2.
    There is no carry created when Y=2.
    Looking at the remaining digits, we see that 2X=16 and so X=8.
    Checking, if X=8 and Y=2 and Z=5, we obtain 825+825+25 which equals 1675, as required.
    Therefore, X+Y+Z=8+2+5=15.

    Answer: (B)

  14. Since Igor is shorter than Jie, then Igor cannot be the tallest.
    Since Faye is taller than Goa, then Goa cannot be the tallest.
    Since Jie is taller than Faye, then Faye cannot be the tallest.
    Since Han is shorter than Goa, then Han cannot be the tallest.
    The only person of the five who has not been eliminated is Jie, who must thus be the tallest.

    Answer: (E)

  15. Since there are 32 red marbles in the bag and the ratio of red marbles to blue marbles is 4:7, then there are 74(32)=56 blue marbles in the bag.
    Since there are 56 blue marbles in the bag and the ratio of blue marbles to purple marbles is 2:3, then there are 32(56)=84 purple marbles in the bag.
    Since the bag contains only red, blue and purple marbles, then there are 32+56+84=172 marbles in the bag.

    Answer: (E)

  16. Since x+2y=30, then x5+2y3+2y5+x3=x5+2y5+x3+2y3=15x+15(2y)+13x+13(2y)=15(x+2y)+13(x+2y)=15(30)+13(30)=6+10=16

    Answer: (B)

  17. First, we factor 1230 as a product of prime numbers: 1230=2×615=2×5×123=2×5×3×41 We are looking for three positive integers r,s,t with r×s×t=1230 and whose sum is as small as possible. We note that all of the possibilities given for the smallest possible value of r+s+t are less than 60.
    Since 41 is a prime factor of 1230, then one of r,s,t must be a multiple of 41.
    Since all of the possibilities given for the minimum sum r+s+t are less than 60 and the second smallest multiple of 41 is 82, then the multiple of 41 in the list r,s,t must be 41 itself.
    Thus, we can let t=41.
    Now we are looking for positive integers r and s with r×s=2×3×5=30 and whose sum r+s is as small as possible. (Since t is now fixed, then minimizing r+s+t is equivalent to minimizing r+s.)
    The possible pairs of values for r and s in some order are 1,30 and 2,15 and 3,10 and 5,6.
    The pair with the smallest sum is 5,6.
    Therefore, we set r=5 and s=6. This gives r+s+t=5+6+41=52.

    Answer: (B)

  18. Since 17 is positive and 176n, then 6n is positive and so n is positive.
    Since 17=642 and 14=624, then the given inequality is equivalent to 6426n624.
    Since the fractions are all positive and n>0, then this is true when 24n42. (If two fractions have the same numerator, then the larger fraction has a smaller denominator.)
    Since n is an integer, then there are 4224+1=19 possible values for n. (We could count the integers from 24 to 42, inclusive, to confirm this.)

    Answer: (C)

  19. We start by drawing a graph that includes the point (1,1), the lines with slopes 14 and 54 that pass through this point, the vertical line with equation x=5, and the horizontal line y=1 (which passes through (1,1) and is perpendicular to the vertical line with equation x=5).
    We label the various points of intersection (A, B, C) as shown.

    From top to bottom, the points are labelled C B A.

    We want to determine the area of PBC.
    Since P has coordinates (1,1) and A has coordinates (5,1), then PA=4.
    Since the slope of PB is 14 and PA=4, then thinking about slope as “rise over run”, we see that AB=1.
    Since the slope of PC is 54 and PA=4, then AC=5.
    Since AC=5 and AB=1, then BC=ACAB=51=4.
    We can view PBC as having base BC and perpendicular height PA. (This is because the length of PA is the perpendicular distance from the line through B and C to the point P.)
    Therefore, the area of this triangle is 12(4)(4) which equals 8.
    Alternatively, we could note that the area of PBC equals the area of PAC minus the area of PAB.
    PAC has base PA=4 and height AC=5 and so has area 12(4)(5)=10.
    PAB has base PA=4 and height AB=1 and so has area 12(4)(1)=2.
    Thus, the area of PBC is 102=8.

    Answer: (C)

  20. Since there are 60 seconds in 1 minute, then t seconds is equivalent to t60 minutes.
    Since there are 60 minutes in 1 hour, then t60 minutes is equivalent to t60×60 hours ort3600 hours.
    Consider the distances that Car X and Car Y travel between the instant when the front of Car Y is lined up with the back of Car X and the instant when the back of Car Y is lined up with the front of Car X.
    Since the length of Car X is 5 m and the length of Car Y is 6 m, then during this interval of time, Car Y travels 5+6=11 m farther than Car X. (The front of Car Y must, in some sense, travel all of the way along the length of the Car X and be 6 m ahead of the front of Car X so that the back of Car Y is lined up with the front of Car X.)
    Since there are 1000 metres in 1 km, then 11 m is equivalent to 0.011 km.
    Since Car X travels at 90 km/h, then in t3600 hours, Car X travels 90t3600 km.
    Since Car Y travels at 91 km/h, then in t3600 hours, Car Y travels 91t3600 km.
    Therefore, 91t360090t3600=0.011, or t3600=0.011 and so t=3600×0.011=36×1.1=39.6.

    Answer: (A)

  21. We label the remaining unknown entries as a,b,c,d, as shown.
    Now a,b,c,d,x must equal 2,3,4,5,6 in some order, with the restriction that no two integers that differ by 1 may be in squares that share an edge.
    In particular, we see first that a2. Therefore, a can equal 3, 4, 5, or 6.
    If a=3, then neither b nor d can equal 2 or 4.
    Thus, b and d equal 5 and 6 in some order.
    In this case, c cannot be 4 (since one of b and d is 5) so c=2 and so x=4.    

    The left stack is labelled b c x from top to bottom. The right stack is labelled 1 a d from top to bottom.
    If a=4, then neither b nor d can equal 3 or 5.
    Thus, b and d equal 2 and 6 in some order.
    In this case, c cannot equal 3 or 5, since it is adjacent to 2 and 6, so this case is not possible.
    If a=5, then neither b nor d can equal 4 or 6.
    Thus, b and d equal 2 and 3 in some order.
    In this case, c cannot be 4 (since one of b and d is 3) so c=6 and so x=4.
    If a=6, then neither b nor d can equal 5.
    Thus, b and d equal two of 2, 3 or 4 in some order.
    If b and d are 2 and 4, then c cannot be 3 or 5 (the remaining numbers), so there is no solution.
    If b and d are 3 and 4, then c cannot be 2 or 5 (the remaining numbers), so there is no solution.
    If b and d are 2 and 3, then we can have c=5, in which case we must have x=4, which is not possible.
    Having considered all possible cases, we see that there is only one possible value for x, namely x=4.

    Answer: (A)

  22. Suppose that the height of the top left rectangle is 2x.
    Since square PQRS has side length 42, then the bottom rectangle has height 422x.
    Since the width of the bottom rectangle is 42 (the side length of the square), then the perimeter of the bottom rectangle is 2(42)+2(422x)=1684x.
    Suppose that the width of the top left rectangle is y.
    Since each of the small rectangles has the same perimeter, then 2(2x)+2y=1684x. (That is, the perimeter of the top left rectangle equals the perimeter of the bottom rectangle.)
    Thus, 2y=1688x or y=844x.
    Since the side length of the square is 42, then the width of the top right rectangle is 42(844x)=4x42.
    Since the two rectangles on the top right have the same perimeter and the same width, then they must have the same height, which is equal to half of the height of the top left rectangle.
    Thus, the height of each of the two rectangles on the right is x.
    Finally, the perimeter of the top right rectangle must equal also equal 1684x.
    Thus, 2(4x42)+2x=1684x or 8x84+2x=1684x and so 14x=252 or x=18.
    This tells us that the shaded rectangle has dimensions x=18 by 4x42=4(18)42=30 and so its area is 18×30=540.
       

    (We can check by substitution that the resulting rectangles in the original diagram are 6×42, 36×12 and 30×18, which all have the same perimeter.)

    Answer: (E)

  23. To solve this problem, we need to use two facts about a triangle with positive area and side lengths abc:

    Suppose that the unknown side length x of the obtuse triangle is the longest side length; that is, suppose that 1017x.
    Here, we must have 10+17>x and 102+172<x2.
    From the first inequality, x<27. Since x is an integer, then x26.
    From the second inequality, x2>389 and so x>389. Since 38919.72 and x is an integer, then x20. This gives the possible values x=20,21,22,23,24,25,26
    Suppose that the unknown side length x of the obtuse triangle is not the longest side length. That is, x17. (We know that 1017 as well. Note also that the relative size of 10 and x is not important.)
    Here, we must have x+10>17 and 102+x2<172.
    From the first inequality, x>7. Since x is an integer, then x8.
    From the second inequality, x2<189 and so x<189. Since 18913.75 and x is an integer, then x13. This gives the possible values x=8,9,10,11,12,13.
    Therefore, the possible values of x are 8,9,10,11,12,13,20,21,22,23,24,25,26.
    The sum of these possible values is 224.
    We still need to justify the second fact. We show that if the triangle is obtuse, then a2+b2<c2. The second half of the fact can be shown in a similar way. While we could use the cosine law to justify this statement, we show this using angles and side lengths:

    Consider ABC with ACB obtuse and with AB=c, AC=b and BC=a.
    At C, draw a perpendicular line segment CD with length b. Draw DB and DA.

    Since BCD is right-angled at C, then BD=BC2+CD2=a2+b2, by the Pythagorean Theorem.
    Now ACD is isosceles with CA=CD.
    Thus, CDA=CAD.
    But BDA>CDA=CAD>BAD.
    Since BDA>BAD, then in BDA, we have BA>BD.
    This means that c=BA>BD=a2+b2 and so c2>a2+b2, as required.

    Answer: (E)

  24. We think of each allowable sequence of moves as a string of X’s, Y’s and Z’s. For example, the string ZZYXZ would represent moving Z one space to the right, then Z, then Y, then X, then Z.
    For each triple (x,y,z) of integers with 0x3 and 0y3 and 0z3, we define S(x,y,z) to be the number of sequences of moves which result in X moving x spaces to the right and Y moving y spaces to the right and Z moving z spaces to the right.
    For example, S(1,0,0)=0 and S(0,1,0)=0 since X and Y are not allowable sequences, and S(0,0,1)=1 since Z is allowable and is only sequence of 1 move.
    We want to find S(3,3,3).
    Next, we note that if x>y or y>z or x>z, then S(x,y,z)=0, since any allowable sequence has to have at least as many Z’s as Y’s and at least as many Y’s as X’s, because no coin can jump another coin. In other words, we need to have 0xyz3.
    We now make the key observation that if 0xyz3, then S(x,y,z)=S(x1,y,z)+S(x,y1,z)+S(x,y,z1) where we use the convention that if x=0, then S(x1,y,z)=0, and if y=0, then S(x,y1,z)=0, and if z=0, then S(x,y,z1)=0.
    This rule is true because:

    Using this rule, we can create tables of values of S(x,y,z). We create one table for each of z=1, z=2 and z=3.

    In each table, values of y are marked along the left side and values of x along the top. In each table, there are several 0s in spots where x>y or x>z or y>z.
    z=1
    0 1 2 3
    0 1 0 0 0
    1 1 1 0 0
    2 0 0 0 0
    3 0 0 0 0
    z=2
    0 1 2 3
    0 1 0 0 0
    1 2 3 0 0
    2 2 5 5 0
    3 0 0 0 0
    z=3
    0 1 2 3
    0 1 0 0 0
    1 3 6 0 0
    2 5 16 21 0
    3 5 21 42 42

    The non-zero entries are filled from top to bottom and left to right, noting that each entry is the sum of the entries (if applicable) to the left in the same table (this is S(x1,y,z)), above in the same table (this is S(x,y1,z)), and in the same position in the previous table (this is S(x,y,z1)).
    Also, S(0,0,1)=1 (the sequence Z), S(0,1,1)=1 (the sequence ZY), and S(1,1,1)=1 (the sequence ZYX.
    From these tables, we see that S(3,3,3)=42 and so the number of different sequences is 42.

    Answer: (C)

  25. We proceed using several steps.
    Step 1: Least common multiples
    For each positive integer n3 and positive integer x<n, we define L(n,x) to be the least common multiple of the n2 integers 1,2,3,,x2,x1,x+2,x+3,,n1,n.
    One way to calculate the least common multiple of a list of integers is to determine the prime factorization of each of the integers in the list and then to create the product of the largest of each of the prime powers that occurs among the integers in the list.
    For example, if n=9 and x=6, then L(9,6) is the least common multiple of the five integers 1,2,3,4,5,8,9.
    The prime factorizations of the integers larger than 1 in this list are 2=21, 3=31, 4=22, 5=51, 8=23, 9=32 and so L(9,6)=233251=360. In this case, L(9,6) is not divisible by x+1=7 but is divisible by x=6.
    We note that, for any list of integers, a common multiple, m, of all of the integers in this list is always itself a multiple of their least common multiple l. This is because if p is a prime number and a is a positive integer for which pa is a factor of l, then there must be an integer in the list that is a multiple of pa. For m to be a common multiple of every number in the list, pa must also be a factor of m. Since this is true for every prime power pa that is a factor of l, then m is a multiple of l.
    Step 2: Connection between m and L(n,x)
    We show that n is a Nella number with corresponding x exactly when L(n,x) is not divisible by x or x+1.
    Suppose that n is a Nella number with corresponding x and m.
    Then m must be divisible by each of 1,2,3,,x2,x1,x+2,x+3,,n1,n.
    From Step 1, m is a multiple of L(n,x).
    Since m is a multiple of L(n,x) and m is not divisible by x or x+1, then L(n,x) is not either.
    Since L(n,x) is divisible by every other positive integer from between 1 and n, inclusive, then L(n,x) satisfies the required conditions for m in the definition of a Nella number.
    Also, if n and x are positive integers with n3 and x<n and L(n,x) has the two required conditions in the definition of a Nella number, then n is indeed a Nella number.
    Putting this together, n is a Nella number with corresponding x exactly when L(n,x) is not divisible by x or x+1.
    Step 3: Re-statement of problem
    Based on Step 2, we now want to find all positive integers n with 50n2017 for which there exists a positive integer x with x<n with the property that L(n,x) is not divisible by x and x+1.
    Step 4: If n is a Nella number with corresponding x, then x and x+1 are both prime powers
    Suppose that n is a Nella number with corresponding x.
    Further, suppose that x is not a prime power.
    Then x=pab for some prime number p, positive integer a and positive integer b>1 that is not divisible by p.
    In this case, pa<x and b<x and so pa and b are both in the list 1,2,3,,x2,x1, which means that L(n,x) is a multiple of both pa and b and so is a multiple of pab=x. (It is important here that pa and b have no common prime factors.) This is a contradiction.
    This means that if n is a Nella number, then x is a prime power.
    Similarly, x+1 must also be a prime power. To see this, we use the same argument with the additional observation that, because x+1 and x are consecutive, they cannot have any common divisor larger than 1 and so if x+1=pcd, then x cannot equal pc or d and therefore both pc and d are indeed in the list 1,2,3,,x1.
    Step 5: Further analysis of x and x+1
    Suppose that n is a Nella number with corresponding x.
    From Step 4, both x and x+1 are prime powers.
    Since x and x+1 are consecutive, then one is even and one is odd.
    In other words, one of x and x+1 is a power of 2 and the other is a power of an odd prime.
    Furthermore, we know that L(n,x) is not divisible by x or x+1.
    This means that the list x+2,x+3,,n1,n cannot contain a multiple of x or x+1.
    This means that x<n<2x and x+1n<2(x+1), because the “next” multiple of x is 2x and the next multiple of x+1 is 2(x+1).
    Since one of x and x+1 is a power of 2, then 2 times this power of 2 (that is, 2x or 2(x+1)) is the next power of 2, and so the inequalities tell us that n is smaller than the next power of 2, and so x or x+1 has to be the largest power of 2 less than (or less than or equal to, respectively) n.
    Step 6: Second re-statement of problem
    We want to find all positive integers n with 50n2017 for which there exists a positive integer x with x<n with the property that L(n,x) is not divisible by x and x+1.
    From Step 5, we know that either x is the largest power of 2 less than n or x+1 is the largest power of 2 less than or equal to n.
    This means that we want to find all positive integers n with 50n2017 for which at least one of the following two statements is true for some positive integer x<n:

    Step 7: Determining Nella numbers
    We make two separate tables, one where x<n is a power of 2 and one where x+1n is a power of 2. In each case, we determine whether x+1 or x is also a prime power.

    Range of n Largest power of 2 less than n x x+1 Prime power? Nella?
    50n64 32 32 33 No (33=311) No
    65n128 64 64 65 No (65=513) No
    129n256 128 128 129 No (129=343) No
    257n512 256 256 257 Yes See below
    513n1024 512 512 513 No (513=3171) No
    1025n2017 1024 1024 1025 No (1025=5205) No

    Note that 257 is a prime number since it is not divisible by any prime number less than 25716.03. (These primes are 2,3,5,7,11,13.)
    For each n with 257n511, L(n,256) is not divisible by 256 or 257, so each of these n (there are 511257+1=255 of them) is a Nella number.
    For n=512, L(n,256) is divisible by 256 (since 512 is divisible by 256), so n=512 is not a Nella number as this is the only possible candidate for x in this case.

    Range of n Largest power of 2 at most n x+1 x Prime power? Nella?
    50n63 32 32 31 Yes See below
    64n127 64 64 63 No (63=321) No
    128n255 128 128 127 Yes See below
    256n511 256 256 255 No (255=551) No
    512n1023 512 512 511 No (511=773) No
    1024n2017 1024 1024 1023 No (1023=3341) No

    Note that 31 and 127 are prime.
    For each n with 50n61, L(n,31) is not divisible by 31 or 32, so each of these n (there are 6150+1=12 of them) is a Nella number.
    For n=62,63, L(n,31) is divisible by 31 (since 62 is divisible by 31), so neither is a Nella number as this is the only possible candidate for x in this case.
    For each n with 128n253, L(n,127) is not divisible by 127 or 128, so each of these n (there are 253128+1=126 of them) is a Nella number.
    For n=254,255, L(n,127) is divisible by 127 (since 254 is divisible by 127), so neither is a Nella number as this is the only possible candidate for x in this case.
    Step 8: Final tally
    From the work above, there are 255+12+126=393 Nella numbers n with 50n2017.

    Answer: (A)