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2019 Cayley Contest
Solutions
(Grade 10)

Tuesday, February 26, 2019
(in North America and South America)

Wednesday, February 27, 2019
(outside of North American and South America)

©2018 University of Waterloo


  1. Evaluating, 2×0+19=0+19=8.

    Answer: (A)

  2. Kai was born 25 years before 2020 and so was born in the year 202025=1995.

    Answer: (C)

  3. Since 38% of students received a muffin, then 100%38%=62% of students did not receive a muffin.
    Alternatively, using the percentages of students who received yogurt, fruit or a granola bar, we see that 10%+27%+25%=62% did not receive a muffin.

    Answer: (D)

  4. Re-arranging the order of the numbers being multiplied, (2×13)×(3×12)=2×12×3×13=(2×12)×(3×13)=1×1=1

    Answer: (C)

  5. Since 10d+8=528, then 10d=520 and so 10d5=5205 which gives 2d=104.

    Answer: (A)

  6. The line with equation y=x+4 has a y-intercept of 4.
    When the line is translated 6 units downwards, all points on the line are translated 6 units down.
    This moves the y-intercept from 4 to 46=2.

    Answer: (E)

  7. Since the average of 2, x and 10 is x, then 2+x+103=x.
    Multiplying by 3, we obtain 2+x+10=3x.
    Re-arranging, we obtain x+12=3x and then 2x=12 which gives x=6.

    Answer: (E)

  8. To get from P to A, Alain travels 5 units right and 4 units up, for a total distance of 5+4=9 units. (Any path from P to A that only moves right and up will have this same length.)
    To get from P to B, Alain travels 6 units right and 2 units up, for a total distance of 8 units.
    To get from P to C, Alain travels 3 units right and 3 units up, for a total distance of 6 units.
    To get from P to D, Alain travels 5 units right and 1 unit up, for a total distance of 6 units.
    To get from P to E, Alain travels 1 unit right and 4 units up, for a total distance of 5 units.
    Therefore, the shortest path is from P to E.

    Answer: (E)

  9. Since (pq)(qr)(rp)=16, then pqqrrp=16 or ppqqrr=16 which gives p2q2r2=16.
    Thus, (pqr)2=16 and so pqr=±4.
    Using the given answers, pqr is positive and so pqr=4.

    Answer: (C)

  10. Matilda and Ellie each take 12 of the wall.
    Matilda paints 12 of her half, or 12×12=14 of the entire wall.
    Ellie paints 13 of her half, or 13×12=16 of the entire wall.
    Therefore, 14+16=312+212=512 of the wall is painted red.

    Answer: (A)

  11. We let the values in the two unlabelled circles be y and z, as shown.

    A description of the diagram follows.


    From the given rules, y+600=1119 and so y=519.
    Also, z+1119=2019 and so z=900.
    Finally, x+y=z and so x=zy=900519=381.

    Answer: (B)

  12. Join P to R.

    A regular pentagon with the vertices P, Q, R, S, and T. There is a line connecting vertices P and R.

    Since PQRST is a regular pentagon, then PQR=QRS=108.
    Since PQ=QR, then PQR is isosceles with QPR=QRP.
    Since PQR=108, then PQR+QPR+QRP=180108+2QRP=1802QRP=72QRP=36 Since QRS=108, then PRS=QRSQRP=10836=72.

    Answer: (A)

  13. From the ones column, we see that 3+2+q must have a ones digit of 2.
    Since q is between 1 and 9, inclusive, then 3+2+q is beween 6 and 14.
    Since its ones digit is 2, then 3+2+q=12 and so q=7.
    This also means that there is a carry of 1 into the tens column.
    From the tens column, we see that 1+6+p+8 must have a ones digit of 4.
    Since p is between 1 and 9, inclusive, then 1+6+p+8 is beween 16 and 24.
    Since its ones digit is 4, then 1+6+p+8=24 and so p=9.
    This also means that there is a carry of 2 into the hundreds column.
    From the hundreds column, we see that 2+n+7+5 must have a ones digit of 0.
    Since n is between 1 and 9, inclusive, then 2+n+7+5 is beween 15 and 23.
    Since its ones digit is 0, then 2+n+7+5=20 and so n=6.
    This also means that there is a carry of 2 into the thousands column.
    This means that m=2.
    This gives 663792+ 5872042 Thus, we have m+n+p+q=2+6+9+7=24.

    Answer: (B)

  14. Each letter A, B, C, D, E appears exactly once in each column and each row.
    The entry in the first column, second row cannot be A or E or B (the entries already present in that column) and cannot be C or A (the entries already present in that row).
    Therefore, the entry in the first column, second row must be D.
    This means that the entry in the first column, fourth row must be C.
    The entry in the fifth column, second row cannot be D or C or A or E and so must be B.
    This means that the entry in the second column, second row must be E.
    Using similar arguments, the entries in the first row, third and fourth columns must be D and B, respectively.
    This means that the entry in the second column, first row must be C.
    Using similar arguments, the entries in the fifth row, second column must be A.
    Also, the entry in the third row, second column must be D.
    This means that the letter that goes in the square marked with must be B.
    We can complete the grid as follows: ACDBEDECABEDBCACBAEDBAEDC

    Answer: (B)

  15. The slope of line segment PR is 2104 which equals 14.
    Since QPR=90, then PQ and PR are perpendicular.
    This means that the slopes of PQ and PR have a product of 1.
    Since the slope of PR is 14, then the slope of PQ is 4.
    Since the “run” of PQ is 20=2, then the “rise” of PQ must be 4×2=8.
    Thus, s2=8 and so s=10.

    Answer: (C)

  16. Suppose that there are p people behind Kaukab.
    This means that there are 2p people ahead of her.
    Including Kaukab, the total number of people in line is n=p+2p+1=3p+1, which is one more than a multiple of 3.
    Of the given choices (23, 20, 24, 21, 25), the only one that is one more than a multiple of 3 is 25, which equals 3×8+1.
    Therefore, a possible value for n is 25.

    Answer: (E)

  17. Consider the triangular-based prism on the front of the rectangular prism.
    This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back.

    A regular pentagon PQRST with a line from vertices P to R.

    The rectangle on the front measures 3×12 and so has area 36.
    The rectangle on the left measures 3×5 and so has area 15.
    The triangles on the top and bottom each are right-angled and have legs of length 5 and 12. This means that each has area 12×12×5=30.
    The rectangle on the back has height 3. The length of this rectangle is the length of the diagonal of the bottom face of the rectangular prism. By the Pythagorean Theorem, this length is 52+122=25+144=169=13. Thus, this rectangle is 3×13 and so has area 39.
    In total, the surface area of the triangular prism is thus 36+15+2×30+39=150.

    Answer: (D)

  18. André runs for 10 seconds at a speed of y m/s.
    Therefore, André runs 10y m.
    Carl runs for 20 seconds before André starts to run and then 10 seconds while André is running. Thus, Carl runs for 30 seconds.
    Since Carl runs at a speed of x m/s, then Carl runs 30x m.
    Since André and Carl run the same distance, then 30x m=10y m, which means that yx=3.
    Thus, y:x=3:1.

    Answer: (D)

  19. Using exponent laws, the expression 2x+y2xy=2(x+y)(xy)=22y.
    Since x and y are positive integers with xy=6, then the possible values of y are the positive divisors of 6, namely 1, 2, 3, or 6. (These correspond to x=6,3,2,1.)
    The corresponding values of 22y are 22=4, 24=16, 26=64, and 212=4096.
    Therefore, the sum of the possible values of 2x+y2xy is 4+16+64+4096=4180.

    Answer: (A)

  20. Let the radii of the circles with centres X, Y and Z be x, y and z, respectively.
    The distance between the centres of two touching circles equals the sum of the radii of these circles.
    Therefore, XY=x+y which means that x+y=30.
    Also, XZ=x+z which gives x+z=40 and YZ=y+z which gives y+z=20.
    Adding these three equations, we obtain (x+y)+(x+z)+(y+z)=30+40+20 and so 2x+2y+2z=90 or x+y+z=45.
    Since x+y=30 and x+y+z=45, then 30+z=45 and so z=15.
    Since y+z=20, then y=20z=5.
    Since x+z=40, then x=40z=25.
    Knowing the radii of the circles will allow us to calculate the dimensions of the rectangle.
    The height of rectangle PQRS equals the “height” of the circle with centre X, which is length of the diameter of the circle, or 2x.
    Thus, the height of rectangle PQRS equals 50.
    To calculate the width of rectangle PQRS, we join X to the points of tangency (that is, the points where the circle touches the rectangle) T and U on PS and SR, respectively, Z to the points of tangency V and W on SR and QR, respectively, and draw a perpendicular from Z to H on XU.
    Since radii are perpendicular to tangents at points of tangency, then XT, XU, ZV, and ZW are perpendicular to the sides of the rectangle.

    Each of XTSU and ZWRV has three right angles, and so must have four right angles and so are rectangles.
    Thus, SU=TX=25 (the radius of the circle with centre X) and VR=ZW=15 (the radius of the circle with centre Z).
    By a similar argument, HUVZ is also a rectangle.
    Thus, UV=HZ and HU=ZV=15.
    Since XH=XUHU, then XH=10.
    By the Pythagorean Theorem, HZ=XZ2XH2=402102=1500=1015 and so UV=1015.
    This means that SR=SU+UV+VR=25+1015+15=40+1015.
    Therefore, the area of rectangle PQRS is 50×(40+1015)=2000+500153936.5.
    Of the given choices, this answer is closest to (E) 3950.

    Answer: (E)

  21. Solution 1
    We start with the ones digits.
    Since 4×4=16, then T=6 and we carry 1 to the tens column.
    Looking at the tens column, since 4×6+1=25, then S=5 and we carry 2 to the hundreds column.
    Looking at the hundreds column, since 4×5+2=22, then R=2 and we carry 2 to the thousands column.
    Looking at the thousands column, since 4×2+2=10, then Q=0 and we carry 1 to the ten thousands column.
    Looking at the ten thousands column, since 4×0+1=1, then P=1 and we carry 0 to the hundred thousands column.
    Looking at the hundred thousands column, 4×1+0=4, as expected.
    This gives the following completed multiplication: 102564×4410256 Finally, P+Q+R+S+T=1+0+2+5+6=14.

    Solution 2
    Let x be the five-digit integer with digits PQRST.
    This means that PQRST0=10x and so PQRST4=10x+4.
    Also, 4PQRST=400000+PQRST=4000000+x.
    From the given multiplication, 4(10x+4)=400000+x which gives 40x+16=400000+x or 39x=399984.
    Thus, x=39998439=10256.
    Since PQRST=10256, then P+Q+R+S+T=1+0+2+5+6=14.

    Answer: (A)

  22. Here is one way in which the seven friends can ride on four buses so that the seven restrictions are satisfied:

    Bus 1 Bus 2 Bus 3 Bus 4
    Abu Bai Don Gia
    Cha Fan Eva

    At least 3 buses are needed because of the groups of 3 friends who must all be on different buses.
    We will now show that it is impossible for the 7 friends to travel on only 3 buses.
    Suppose that the seven friends could be put on 3 buses.
    Since Abu, Bai and Don are on 3 different buses, then we assign them to three buses that we can call Bus 1, Bus 2 and Bus 3, respectively. (See the table below.)
    Since Abu, Bai and Eva are on 3 different buses, then Eva must be on Bus 3.
    Since Cha and Bai are on 2 different buses and Cha and Eva are on 2 different buses, then Cha cannot be on Bus 2 or Bus 3, so Cha is on Bus 1.
    So far, this gives

    Bus 1 Bus 2 Bus 3
    Abu Bai Don
    Cha Eva

    The remaining two friends are Fan and Gia.
    Since Fan, Cha and Gia are on 3 different buses, then neither Fan nor Gia is on Bus 1.
    Since Don, Gia and Fan are on 3 different buses, then neither Fan nor Gia is on Bus 3.
    Since Gia and Fan are on separate buses, they cannot both be on Bus 2, which means that the seven friends cannot be on 3 buses only.
    Therefore, the minimum number of buses needed is 4.

    Answer: (B)

  23. Since the wheel turns at a constant speed, then the percentage of time when a shaded part of the wheel touches a shaded part of the path will equal the percentange of the total length of the path where there is “shaded on shaded” contact.
    Since the wheel has radius 2 m, then its circumference is 2π×2 m which equals 4π m.
    Since the wheel is divided into four quarters, then the portion of the circumference taken by each quarter is π m.
    We call the left-hand end of the path 0 m.
    As the wheel rotates once, the first shaded section of the wheel touches the path between 0 m and π3.14 m.
    As the wheel continues to rotate, the second shaded section of the wheel touches the path between 2π6.28 m and 3π9.42 m.
    While the wheel makes 3 complete rotations, a shaded quarter will be in contact with the path over 6 intervals (2 intervals per rotation).
    The path is shaded for 1 m starting at each odd multiple of 1 m, and unshaded for 1 m starting at each even multiple of 1 m.
    We make a chart of the sections where shaded quarters touch the path and the parts of these intervals that are shaded:

    Beginning of quarter (m) End of quarter (m) Shaded parts of path (m)
    0 π3.14 1 to 2; 3 to π
    2π6.28 3π9.42 7 to 8; 9 to 3π
    4π12.57 5π15.71 13 to 14; 15 to 5π
    6π18.85 7π21.99 19 to 20; 21 to 7π
    8π25.13 9π28.27 8π to 26; 27 to 28
    10π31.42 11π34.56 10π to 32; 33 to 34

    Therefore, the total length of “shaded on shaded”, in metres, is 1+(π3)+1+(3π9)+1+(5π15)+1+(7π21)+(268π)+1+(3210π)+1 which equals (162π) m.
    The total length of the path along which the wheel rolls is 3×4π m or 12π m.
    This means that the required percentage of time equals (162π) m12π m×100%25.8%.
    Of the given choices, this is closest to 26%, or choice (E).

    Answer: (E)

  24. We let A be the set {2,3,4,5,6,7,8,9}.
    First, we note that the integer s that Roberta chooses is of the form s=11m for some integer m from the set A, and the integer t that Roberta chooses is of the form t=101n for some integer n from the set A.
    This means that the product rst is equal to r(11m)(101n)=11×101×rmn where each of r,m,n comes from the set A.
    This means that the number of possible values for rst is equal to the number of possible values of rmn, and so we count the number of possible values of rst by counting the number of possible values of rmn.
    We note that A contains only one multiple of 5 and one multiple of 7. Furthermore, these multiples include only one factor of 5 and 7 each, respectively.
    We count the number of possible values for rmn by considering the different possibilities for the number of factors of 5 and 7 in rmn.
    Let y be the number of factors of 5 in rmn and let z be the number of factors of 7 in rmn. Note that since each of r, m and n includes at most one factor of 5 and at most one factor of 7 and each of r, m and n cannot contain both a factor of 5 and a factor of 7, then y+z is at most 3.
    Case 1: y=3
    If rmn includes 3 factors of 5, then r=m=n=5 and so rmn=53.
    This means that there is only one possible value for rmn.
    Case 2: z=3
    Here, it must be the case that rmn=73 and so there is only one possible value for rmn.
    Case 3: y=2 and z=1
    Here, it must be the case that two of r,m,n are 5 and the other is 7.
    In other words, rmn must equal 52×7.
    This means that there is only one possible value for rmn.
    Case 4: y=1 and z=2
    Here, rmn must equal 5×72.
    This means that there is only one possible value for rmn.
    Case 5: y=2 and z=0
    Here, two of r,m,n must equal 5 and the third cannot be 5 or 7.
    This means that the possible values for the third of these are 2,3,4,6,8,9.
    This means that there are 6 possible values for rmn in this case.
    Case 6: y=0 and z=2
    As in Case 5, there are 6 possible values for rmn.
    Case 7: y=1 and z=1
    Here, one of r,m,n equals 5, one equals 7, and the other must be one of 2,3,4,6,8,9.
    This means that there are 6 possible values for rmn in this case.
    Case 8: y=1 and z=0
    Here, one of r,m,n equals 5 and none of these equal 7. Suppose that r=5.
    Each of m and n equals one of 2,3,4,6,8,9.
    We make a multiplication table to determine the possible values of mn: ×234689246812161836912182427481216243236612182436485481624324864729182736547281 The different values in this table are 4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81 of which there are 16.
    Therefore, there are 16 possible values of rmn in this case.
    Case 9: y=0 and z=1
    As in Case 8, there are 16 possible values of rmn.
    Case 10: y=0 and z=0
    Here, none of r,m,n is 5 or 7.
    This means that each of r,m,n equals one of 2,3,4,6,8,9.
    This means that the only possible prime factors of rmn are 2 and 3.
    Each of 2,3,4,6,8,9 includes at most 2 factors of 3 and only 9 includes 2 factors of 3.
    This means that rmn contains at most 6 factors of 3.
    Let w be the number of factors of 3 in rmn.

    In total, the number of possible values of rmn (and hence of rst) is 2×1+2×1+2×6+6+2×16+(1+2+4+5+6+6+7) which equals 85.

    Answer: (A)

  25. Suppose that PQ=a, PS=b and PU=c.
    Since PQRSTUVW is a rectangular prism, then QR=PS=b and ST=QV=PU=c.
    By the Pythagorean Theorem, PR2=PQ2+QR2 and so 18672=a2+b2.
    By the Pythagorean Theorem, PV2=PQ2+QV2 and so 20192=a2+c2.
    By the Pythagorean Theorem, PT2=PS2+ST2 and so x2=b2+c2.
    Adding these three equations, we obtain 18672+20192+x2=(a2+b2)+(a2+c2)+(b2+c2)18672+20192+x2=2a2+2b2+2c2a2+b2+c2=18672+20192+x22 Since a2+b2=18672, then c2=(a2+b2+c2)(a2+b2)=18672+20192+x2218672=18672+20192+x22 Since a2+c2=20192, then b2=(a2+b2+c2)(a2+c2)=18672+20192+x2220192=1867220192+x22 Since b2+c2=x2, then a2=(a2+b2+c2)(b2+c2)=18672+20192+x22x2=18672+20192x22 Since a, b and c are edge lengths of the prism, then a,b,c>0.
    Since a2>0, then 18672+20192x22>0 and so 18672+20192x2>0 or x2<20192+18672.
    Since b2>0, then 1867220192+x22>0 and so 1867220192+x2>0 or x2>2019218672.
    Since c2>0, then 18672+20192+x22>0 and so 18672+20192+x2>0 which means that x2>1867220192.
    Since the right side of this last inequality is negative and the left side is non-negative, then this inequality is always true.
    Therefore, it must be true that 2019218672<x2<20192+18672.
    Since all three parts of this inequality are positive, then 2019218672<x<20192+18672.
    Since 2019218672768.55 and 20192+186722749.92 and x is an integer, then 769x2749.
    The number of integers x in this range is 2749769+1=1981.
    Every such value of x gives positive values for a2, b2 and c2 and so positive values for a, b and c, and so a rectangular prism PQRSTUVW with the correct lengths of face diagonals.
    Therefore, there are 1981 such integers x.

    Answer: (E)