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2019 Pascal Contest
Solutions
(Grade 9)

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, 2×3+2×3=6+6=12.

    Answer: (D)

  2. Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square.
    Thus, the side length of a square with perimeter 28 is 28÷4=7.

    Answer: (E)

  3. In the diagram, there are 9 hexagons of which 5 are shaded.
    Therefore, the fraction of all of the hexagons that are shaded is 59.

    Answer: (B)

  4. 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)

  5. We know that 12=0.5. Since 490.44 is less than 12=0.5, then 4 cannot be placed in the box. (No integer smaller than 4 can be placed in the box either.) Since 590.56 is greater than 12=0.5, then the smallest integer that can be placed in the box is 5.

    Answer: (D)

  6. Since 4x+14=8x48, then 14+48=8x4x or 62=4x.
    Dividing both sides of this equation by 2, we obtain 4x2=622 which gives 2x=31.

    Answer: (B)

  7. The segment of the number line between 3 and 33 has length 333=30.
    Since this segment is divided into six equal parts, then each part has length 30÷6=5.
    The segment PS is made up of 3 of these equal parts, and so has length 3×5=15.
    The segment TV is made up of 2 of these equal parts, and so has length 2×5=10.
    Thus, the sum of the lengths of PS and TV is 15+10 or 25.

    Answer: (A)

  8. Since 2019 is larger than 1 and smaller than 2, and 20×19=380, then 2019<20×19<2019.
    We note that 1920>1020>10000 and 2019>1019>10000.
    This means that both 1920 and 2019 are greater than 2019.
    In other words, of the five numbers 1920,2019,2019,2019,20×19, the third largest is 2019.
    Since the list contains 5 numbers, then its median is the third largest number, which is 2019.
    (Note that it does not matter whether 1920 is greater than or less than 2019.)

    Answer: (D)

  9. Since the complete angle at the centre of each circle is 360 and the unshaded sector of each circle has central angle 90, then the unshaded sector of each circle represents 90360=14 of its area.
    In other words, each of the circles is 34 shaded.
    There are 12 circles in the diagram.
    Since the radius of each circle is 1, then the area of each circle is π×12 or π.
    Therefore, the total shaded area is 34×12×π or 9π.

    Answer: (D)

  10. Suppose that sixty 1×1×1 cubes are joined face to face in a single row on a table.
    Each of the 60 cubes has its front, top and back faces exposed.
    The leftmost and rightmost cubes also have their left and right faces, respectively, exposed.
    No other faces are exposed.
    Therefore, the number of 1×1 faces that are exposed is 60×3+2 which equals 182.

    Answer: (C)

  11. Using the second row, we see that the sum of the numbers in each row, column and diagonal must be 3.6+3+2.4=9.
    Since the sum of the numbers in the first column must be 9, then the bottom left number must be 92.33.6=95.9=3.1
    Since the sum of the numbers in the top left to bottom right diagonal must be 9, then the bottom right number must be 92.33=95.3=3.7
    Since the sum of the numbers in the bottom row must be 9, then 3.1+x+3.7=9 and so x+6.8=9 or x=96.8=2.2.
    We can complete the magic square as shown: 2.33.82.93.632.43.12.23.7

    Answer: (E)

  12. Since PQX is right-angled at Q, then PXQ=90QPX=9062=28 Since PXQ and SXR are opposite, then SXR=PXQ=28.
    Since RXS is isosceles with RX=SX, then XRS=XSR=y.
    Since the angles in any triangle have a sum of 180, XRS+XSR+SXR=180y+y+28=1802y+28=1802y=152 and so y=76.

    Answer: (C)

  13. Solution 1
    Since p,q,r,s is a list of consecutive integers in increasing order, then q is 1 more than p and r is 1 less than s.
    This means that q+r=(p+1)+(s1)=p+s=109.

    Solution 2
    Since p,q,r,s is a list of consecutive integers in increasing order, then q=p+1, r=p+2, and s=p+3.
    Since p+s=109, then p+p+3=109 and so 2p=106 or p=53.
    This means that q=54 and r=55. Thus, q+r=109.

    Answer: (B)

  14. Since the ratio of the number of skateboards to the number of bicycles was 7:4, then the numbers of skateboards and bicycles can be written in the form 7k and 4k for some positive integer k.
    Since the difference between the numbers of skateboards and bicycles is 12, then 7k4k=12 and so 3k=12 or k=4.
    Therefore, the total number of skateboards and bicycles is 7k+4k=11k=11×4=44.

    Answer: (A)

  15. For Sophie’s average over 5 tests to be 80%, the sum of her marks on the 5 tests must be 5×80%=400%.
    After the first 3 tests, the sum of her marks is 73%+82%+85%=240%.
    Therefore, she will reach her goal as long as the sum of her marks on the two remaining tests is at least 400%240%=160%.
    The sums of the pairs of marks given are (A) 161%, (B) 161%, (C) 162%, (D) 156%, (E) 160%.
    Thus, the pair with which Sophie would not meet her goal is (D).

    Answer: (D)

  16. Solution 1
    Since the result must be the same for any real number x less than 2, we substitute x=4 into each of the five expressions:

    (A) x=4 (B) x+2=2 (C) 12x=2 (D) x2=6 (E) 2x=8

    Therefore, 2x is the expression with the least value when x=4 and thus must always be the expression with the least value.

    Solution 2
    For any real number x, we know that x2 is less than x which is less than x+2.
    Therefore, neither x nor x+2 can be the least of the five values.
    For any negative real number x, the value of 2x will be less than the value of 12x.
    Therefore, 12x cannot be the least of the five values.
    Thus, the least of the five values is either x2 or 2x.
    When x<2, we know that 2x(x2)=x+2<0.
    Since the difference between 2x and x2 is negative, then 2x has the smaller value and so is the least of all five values.

    Answer: (E)

  17. Each of the animals is either striped or spotted, but not both.
    Since there are 100 animals and 62 are spotted, then there are 10062=38 striped animals.
    Each striped animal must have wings or a horn, but not both.
    Since there are 28 striped animals with wings, then there are 3828=10 striped animals with horns.
    Each animal with a horn must be either striped or spotted.
    Since there are 36 animals with horns, then there are 3610=26 spotted animals with horns.

    Answer: (E)

  18. By the Pythagorean Theorem, QT2=QP2+PT2=k2+k2=2k2 Since QT>0, then QT=2k.
    Since QTS is isosceles, then TS=QT=2k.
    By the Pythagorean Theorem, QS2=QT2+TS2=(2k)2+(2k)2=2k2+2k2=4k2 Since QS>0, then QS=2k.
    Since QSR is isosceles, then SR=QS=2k.
    Since QPT is right-angled at P, its area is 12(QP)(PT)=12k2.
    Since QTS is right-angled at T, its area is 12(QT)(TS)=12(2k)(2k)=12(2k2)=k2.
    Since QSR is right-angled at S, its area is 12(QS)(SR)=12(2k)(2k)=2k2.
    Since the sum of the three areas is 56, then 12k2+k2+2k2=56 or 72k2=56 which gives k2=16.
    Since k>0, then k=4.

    Answer: (C)

  19. Since 4 balls are chosen from 6 red balls and 3 green balls, then the 4 balls could include:

    There is only 1 different-looking way to arrange 4 red balls.
    There are 4 different-looking ways to arrange 3 red balls and 1 green ball: the green ball can be in the 1st, 2nd, 3rd, or 4th position.
    There are 6 different-looking ways to arrange 2 red balls and 2 green balls: the red balls can be in the 1st/2nd, 1st/3rd, 1st/4th, 2nd/3rd, 2nd/4th, or 3rd/4th positions.
    There are 4 different-looking ways to arrange 1 red ball and 3 green balls: the red ball can be in the 1st, 2nd, 3rd, or 4th position.
    In total, there are 1+4+6+4=15 different-looking arrangements.

    Answer: (A)

  20. Since the sides of quadrilateral WXYZ are parallel to the diagonals of square PQRS and the diagonals of a square are perpendicular, then the sides of WXYZ are themselves perpendicular.
    This means that quadrilateral WXYZ has four right angles and is thus a rectangle.
    Since the diagram does not change when rotated by 90, 180 or 270, then it must be the case that WX=XY=YZ=YW which means that WXYZ is a square.
    We calculate the area of WXYZ by first calculating the length of WZ.
    Extend KW to meet PQ at T. Join M to T.

    A square with the vertices P, Q, R, and S, starting from the top left corner and going clockwise. Point J is on the side of PQ, point K is on the side of QR, point L is on the side of RS, and the point M is on the side of SP. There is a diamond in the middle of the square with vertices W, X, Y, and Z. Vertex W is on the line JZ, vertex X is on the line KW, vertex Y is on the line LX, vertex Z is on the line MY. Point t is on the side of PJ, and the point M is on the side of PS. There are lines connecting points T, M, Z, and W to form another square.

    Since TK is parallel to diagonal PR, then QTK=QKT=45, which means that TQK is isosceles with QT=QK.
    Since QR=40 and KR=10, then QK=QRKR=30 and so QT=30.
    Since PQ=40, then PT=PQQT=10.
    Since PM=PT=10, then MPT is right-angled and isosceles as well, which means that MT is actually parallel to diagonal SQ. (We did not construct MT with this property, but it turns out to be true.)
    Since the sides of MTWZ are parallel to the diagonals of the square, then MTWZ is also a rectangle, which means that WZ=MT.
    Since PM=PT, then MT=PM2+PT2=102+102=200 by the Pythagorean Theorem.
    Thus, WZ=MT=200 and so the area of square WXYZ equals WZ2 or 200.

    Answer: (B)

  21. The units digit of 52019 is 5.
    This is because the units digit of any power of 5 is 5.
    To see this, we note that the first few powers of 5 are 51=552=2553=12554=62555=312556=15625 The units digit of a product of integers depends only on the units digits of the integers being multiplied. Since 5×5=25 which has a units digit of 5, the units digits remains 5 for every power of 5.
    The units digit of 32019 is 7.
    This is because the units digits of powers of 3 cycle 3,9,7,1,3,9,7,1,.
    To see this, we note that the first few powers of 3 are 31=332=933=2734=8135=24336=729 Since the units digit of a product of integers depends only on the units digits of the integers being multiplied and we multiply by 3 to get from one power to the next, then once a units digit recurs in the sequence of units digits, the following units digits will follow the same pattern.
    Since the units digits of powers of 3 cycle in groups of 4 and 2016 is a multiple of 4, then 32016 has a units digit of 1.
    Moving three additional positions along the sequence, the units digit of 32019 will be 7.
    Since the units digit of 52019 is 5 and the units digit of 32019 is 7, then the units digit of the difference will be 8. (This is the units digit of the difference whenever a smaller integer with units digit 7 is substracted from a larger integer with units digit 5.)

    Answer: (E)

  22. Solution 1
    The smallest integer greater than 2019 that can be formed in this way is formed using the next two largest consecutive integers 20 and 21, giving the four-digit integer 2120.
    The largest such integer is 9998.
    The list of such integers is 2120,2221,2322,,9796,9897,9998 Each pair of consecutive numbers in this list differs by 101 since the number of hundreds increases by 1 and the number of ones increases by 1 between each pair.
    Since the numbers in the list are equally spaced, then their sum will equal the number of numbers in the list times the average number in the list.
    The average number in the list is 2120+99982=121182=6059.
    Since each number in the list is 101 greater than the previous number, then the number of increments of 101 from the first number to the last is 99982120101=7878101=78.
    Since the number of increments is 78, then the number of numbers is 79.
    This means that the sum of the numbers in the list is 79×6059=478661.

    Solution 2
    As in Solution 1, the list of such integers is 2120,2221,2322,,9796,9897,9998.
    If the sum of these integers is S, then S=2120+2221+2322++9796+9897+9998=(2100+2200+2300++9700+9800+9900)+(20+21+22++96+97+98)=100(21+22+23++96+97+98+99)+(20+21+22++96+97+98)=100(21+22+23++96+97+98+99)+(21+22++96+97+98+99)+2099=101(21+22+23++96+97+98+99)79 There are 79 numbers in the list of consecutive integers from 21 to 99, inclusive, and the middle number in this list is 60.
    Therefore, S=101×79×6079=478661.

    Answer: (C)

  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 label 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 π m3.14 m.
    As the wheel continues to rotate, the second shaded section of the wheel touches the path between 2π m6.28 m and 3π m9.42 m.
    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.
    Therefore, the first shaded section touches shaded stripes between 1 m and 2 m, and between 3 m and π m.
    The second shaded section touches shaded stripes between 7 m and 8 m, and between 9 m and 3π m.
    Therefore, the total length of “shaded on shaded” is 1 m+(π3) m+1 m+(3π9) m or (4π10) m.
    The total length of the path along which the wheel rolls is 4π m.
    This means that the required percentage of time equals (4π10) m4π m×100%20.4%.
    Of the given choices, this is closest to 20%, or choice (A).

    Answer: (A)

  24. First, we note that 88663311000 is divisible by 792. (We can check this by division.)
    Therefore, 88663311000000 is also divisible by 792.
    Since 88663311000000 is divisible by 792, then 88663311pqrs48 is divisible by 792 exactly when pqrs48 is divisible by 792. (This comes from the fact that if the difference between two integers is divisible by d, then either both are divisible by d or neither is divisible by d.)
    The smallest integer of the form pqrs48 is 48 (which is “000048”) and the largest integer of the form pqrs48 is 999948.
    Since 999948÷7921262.6, then the multiples of 792 in between 48 and 999948 are the integers of the form 792×n where 1n1262.
    Suppose that 792×n=pqrs48 for some integer n.
    Comparing units digits, we see that the units digit of n must be 4 or 9.
    This means that n=10c+4 or n=10c+9 for some integer c0.
    In the first case, 792(10c+4)=7920c+3168.
    This integer has a units digit of 8.
    For this integer to have a tens digit of 4, we need 2c+6 to have a units digit of 4, which happens exactly when c has units digit 4 or 9.
    This means that c can be 4,9,14,19,24,.
    This means that n can be 44,94,144,194,244,.
    Since 1n1262, then there are 25 possible values of n with units digit 4, because there are 2 values of n between 0 and 100, 2 between 100 and 200, and so on up to 1200, with an additional 1 (namely, 1244) between 1200 and 1262.
    In the second case, 792(10c+9)=7920c+7128.
    This integer has a units digit of 8.
    For this integer to have a tens digit of 4, we need 2c+2 to have units digit 4, which happens exactly when c has units digit 1 or 6.
    This means that c can be 1,6,11,16,21,.
    This means that n can be 19,69,119,169,219,.
    Since 1n1262, then there are again 25 possible values of n with units digit 9.
    In total, there are 25+25=50 14-digit positive integers of the desired form that are divisible by 792.

    Answer: (E)

  25. In ABC, if D is on BC, then Area of ABDArea of ACD=BDCD

    Triangle ABC with vertex A at the top, vertex B at the bottom left, and vertex C at the bottom right. Point D is on the side of BC and there is a line connecting points A and D. The height is also drawn as a dotted line from vertex A.

    This is because ABD and ACD have a common height of h and so Area of ABDArea of ACD=12×BD×h12×CD×h=BDCD Using this in the given diagram with point V lying on side QS of WQS and on side QS of RQS, we see that Area of QVWArea of SVW=QVSV=Area of QVRArea of SVR Combining the first and third parts of this equality, we obtain the equivalent equations 8x+505x+20=8x+325x+11(8x+50)(5x+11)=(8x+32)(5x+20)40x2+88x+250x+550=40x2+160x+160x+640338x+550=320x+64018x=90x=5 Since x=5, then we can calculate and fill in the area of most of the pieces of the diagram:

    Let the area of PTW equal y and the area of QTW equal z.
    We know that Area of QVWArea of SVW=QVSV and so QVSV=9045=2.
    Since V lies on side QS of PQS, then Area of PQVArea of PSV=QVSV=2 and so y+z+9099=2 which gives y+z+90=198 and so y+z=108.
    Finally, looking at points W on side TS of PTS and so side TS of QTS, we get y54=TWWS=z135 and so 135y=54z or 5y=2z or z=52y.
    Therefore, y+52y=108 or 72y=108 and so y=2167=3067.
    Of the given choices, the area of PTW is closest to 31.

    Answer: (E)