CEMC Banner

2024 Cayley Contest
Solutions
(Grade 10)

Wednesday, February 28, 2024
(in North America and South America)

Thursday, February 29, 2024
(outside of North American and South America)

©2023 University of Waterloo


  1. Calculating, 2×0+2×4=0+8=8.

    Answer: (E)

  2. When x=3, we have (5x6x)=(x)=x=3.
    Alternatively, when x=3, we have (5x6x)=(1518)=(3)=3.

    Answer: (B)

  3. Since AE=BF and BE=CF, then AB=AE+BE=BF+CF=BC.
    Therefore, ABC is isosceles with BAC=BCA=70°.
    Since the sum of the angles in ABC is 180°, then ABC=180°BACBCA=180°70°70°=40°

    Answer: (A)

  4. On Friday, the Cayley Comets scored 80% of 90 points.
    This is equal to 80100×90=810×90=8×9=72 points.
    Alternatively, since 80% is equivalent to 0.8, then 80% of 90 is equal to 0.8×90=72.

    Answer: (B)

  5. The volume of a prism is equal to the area of its base times its depth.
    Here, the prism has identical bases with area 400 cm2 and depth 8 cm, and so its volume is 400 cm2×8 cm=3200 cm3.

    Answer: (C)

  6. The percentage of cookies that Lloyd ate that were chocolate chip or oatmeal was 33%+22% which equals 55%.
    This leaves 100%55%=45% of the cookies that were gingerbread or sugar.
    Since Lloyd ate two times as many gingerbread cookies as sugar cookies, then 23 of the 45%, or 30%, were gingerbread cookies.

    Answer: (C)

  7. Simplifying, 16+13=16+26=36=12. Thus, 1x=12 and so x=2.

    Answer: (D)

  8. Since 4=22, then 47=(22)7=214=(27)2, which means that 47 is a perfect square.
    We can check, for example using a calculator, that the square root of each of the other four choices is not an integer, and so each of these four choices cannot be expressed as the square of an integer.

    Answer: (C)

  9. Suppose that the smallest of the five odd integers is x.
    Since consecutive odd integers differ by 2, the other four odd integers are x+2, x+4, x+6, and x+8.
    Therefore, x+(x+2)+(x+4)+(x+6)+(x+8)=125.
    From this, we obtain 5x+20=125 and so 5x=105, which gives x=21.
    Thus, the smallest of the five integers is 21. (This means that the five odd integers are 21, 23, 25, 27, 29.)

    Answer: (C)

  10. When two standard six-sided dice are rolled, there are 6×6=36 possibilities for the pair of numbers that are rolled.
    Of these, the pairs 2×6, 3×4, 4×3, and 6×2 each give 12. (If one of the numbers rolled is 1 or 5, the product cannot be 12.)
    Since there are 4 pairs of possible rolls whose product is 12, the probability that the product is 12 is 436.

    Answer: (B)

  11. Since Arturo has an equal number of $5 bills, of $10 bills, and of $20 bills, then we can divide Arturo’s bills into groups, each of which contains one $5 bill, one $10 bill, and one $20 bill.
    The value of the bills in each group is $5+$10+$20=$35.
    Since the total value of Arturo’s bills is $700, then there are $700$35=20 groups.
    Thus, Arturo has 20 $5 bills.

    Answer: (D)

  12. Since the mass of 2 Exes equals the mass of 29 Wyes, then the mass of 8×2 Exes equals the mass of 8×29 Wyes.
    In other words, the mass of 16 Exes equals the mass of 232 Wyes.
    Since the mass of 1 Zed equals the mass of 16 Exes, then the mass of 1 Zed equals the mass of 232 Wyes.

    Answer: (C)

  13. Draw a perpendicular from D to F on AB.
    Since quadrilateral FDCB has right angles at F, C and B, then it must be a rectangle.
    This means that FB=DC=15 and FD=BC=12.
    Further, AF=ABFB=2015=5.

    Now, AFD is right-angled at F.
    By the Pythagorean Theorem, AD2=AF2+FD2=52+122=25+144=169.
    Since AD>0, then AD=13. (Some might recognize the Pythagorean triple 5-12-13 directly.)
    Thus, the perimeter of ABCD is 20+12+15+13=60.

    Answer: (E)

  14. Since 10 numbers have an average of 87, their sum is 10×87=870.
    When the numbers 51 and 99 are removed, the sum of the remaining 8 numbers is 8705199 or 720.
    The average of these 8 numbers is 7208=90.

    Answer: (A)

  15. The sum of the lengths of the horizontal line segments in Figure 2 is 4x, because the tops of the four small rectangles contribute a total of 2x to their combined perimeter and the bottoms of the four small rectangles contribute a total of 2x to their combined perimeter.
    Similarly, the sum of the lengths of the vertical line segments in Figure 2 is 4y.
    In other words, the sum of the perimeters of the four rectangles in Figure 2 is 4x+4y.
    Since the sum of the perimeters also equals 24, then 4x+4y=24 and so x+y=6.

    Answer: (A)

  16. Since 12×23×34×45××n1n=18 then squaring both sides, we obtain 12×23×34×45××n1n=164 Simplifying the left side, we obtain 1×2×3×4××(n1)2×3×4×5××n=164 or 1×(2×3×4××(n1))(2×3×4××(n1))×n=164 and so 1n=164 which means that n=64.

    Answer: (B)

  17. The integers between 1000 and 9999, inclusive, are all four-digit positive integers of the form abcd.
    We want each of a, b, c, and d to be even.
    There are 4 choices for a, namely 2, 4, 6, 8. (a cannot equal 0.)
    There are 5 choices for each of b, c and d, namely 0, 2, 4, 6, 8.
    The choice of each digit is independent, and so the total number of such integers is 4×5×5×5 or 500.

    Answer: (A)

  18. The line with equation y=3x+5 has slope 3 and y-intercept 5.
    Since the line has y-intercept 5, it passes through (0,5).
    When the line is translated 2 units to the right, its slope does not change and the new line passes through (2,5).
    A line with slope m that passes through the point (x1,y1) has equation yy1=m(xx1). Therefore, the line with slope 3 that passes through (2,5) has equation y5=3(x2) or y5=3x6, which gives y=3x1.

    Alternatively, we could note that when the graph of y=3x+5 is translated 2 units to the right, the equation of the new graph is y=3(x2)+5 or y=3x1.

    Answer: (B)

  19. Since squares DKHG, ELJH and FMCJ have their bases along the same line, then DK, EL and FM are parallel.
    Since DK and EL are parallel, then EDK=FEL.
    Since EKD is right-angled at K and FLE is right-angled at L, then EKD and FLE are similar.

    Since the area of DKHG is 16, then its side length is 16=4.
    Since the area of ELJH is 36, then its side length is 36=6.
    Since EH=6 and KH=4, then EK=2.
    Therefore, EKD has EK=2 and DK=4; in other words, EK:DK=1:2.
    Since FLE is similar to EKD, then FL:LE=1:2.
    Since EL=6, then FL=3. Since LJ=6 and FL=3, then FJ=FL+LJ=9.
    Therefore, the area of square FMCJ is 92 or 81.

    Answer: (D)

  20. Suppose that the length of the race was d m.
    Suppose further that Jiwei finished the first race in t s.
    Since Hari finished in 45 of the time that Jiwei took, then Hari finished in 45t s.
    Since speed equals distance divided by time, then Jiwei’s average speed was dt m/s and Hari’s average speed was d4t/5=54dt m/s.
    For Jiwei to finish in the same time as Hari, Jiwei must increase his average speed from dt m/s to 54dt m/s.
    This is an increase of one-quarter over the original speed, or an increase of 25%. Thus, x=25.

    Answer: (B)

  21. Since the second column includes the number 1, then step (ii) was never used on the second column, otherwise each entry would be at least 2.
    To generate the 1, 3 and 2 in the second column, we thus need to have used step (i) 1 time on row 1, 3 times on row 2, and 2 times on row 3.
    This gives:

    1 1 1
    3 3 3
    2 2 2

    We cannot use step (i) any more times, otherwise the entries in column 2 will increase. Thus, a=1+3+2=6.
    To obtain the final grid from this current grid using only step (ii), we must increase each entry in column 1 by 6 (which means using step (ii) 3 times) and increase each entry in column 3 by 4 (which means using step (ii) 2 times). Thus, b=3+2=5.
    Therefore, a+b=11.

    Answer: 11

  22. Let O be the origin, A be the point with coordinates (20,24), and B be the point with coordinates (4,202).
    The slope of OA is 2420=1.2. The slope of OB is 2024=50.5.

    The line with equation y=mx passes through (0,0).
    If m<0, the line with equation y=mx does not pass through the first quadrant.
    If 0m<1.2, the line with equation y=mx is less steep than OA and so does not intersect line segment AB.
    If m>50.5, the line with equation y=mx is steeper than OB and so does not intersect line segment AB.
    For integers m with 2m50, the line with equation y=mx will intersect the line segment AB.
    There are 49 such integers m.

    Answer: 49

  23. To calculate the shaded area, we add the area of the rectangle and the areas of the four semi-circles, and subtract the area of the larger circle.
    Since the rectangle is 6 by 8, its area is 6×8=48.
    The two semi-circles of diameter 6 together form a complete circle of diameter 6, or radius 3. The combined area of these semi-circles is π×32 or 9π.
    The two semi-circles of diameter 8 together form a complete circle of diameter 8, or radius 4. The combined area of these semi-circles is π×42 or 16π.
    Since the larger circle passes through the four vertices of the rectangle, the diagonal of the rectangle is its diameter. (This is because the diagonal subtends an angle of 90° at each of the other vertices and so is a diameter.)
    The length of the diagonal is 62+82=100=10, and so the radius of the larger circle is 5, and so its area is π×52=25π.
    Finally, this means that the area of the shaded region is 48+9π+16π25π which equals 48.
    The closest integer to 48 is 48.

    Answer: 48

  24. If Rasheeqa walks directly from A to B to C, it takes 2+3=5 minutes.
    We note that Rasheeqa cannot walk from A to C in less than 5 minutes, so t cannot equal 1, 2, 3, or 4.
    Rasheeqa can add increments of 3 minutes to her walk by walking around the circular path that begins and ends at B.
    This means that Rasheeqa’s total time, in minutes, can be each of 5,8,11,14,,98,101.
    Rasheeqa can also walk from A to B to A to B to C. This takes 2+3+2+3=10 minutes.
    Any walk that uses the path from B back to A once must be at least this long.
    Rasheeqa can again add increments of 3 minutes to this 10 minute walk by walking around the circular path that begins and ends at B.
    This means that Rasheeqa’s total time, in minutes, can be each of 10,13,16,19,,97,100,103.
    Rasheeqa can also walk from A to B to A to B to A to B to C. This takes 2+3+2+3+2+3=15 minutes.
    Any walk that uses the path from B back to A twice must be at least this long.
    Rasheeqa can again add increments of 3 minutes to her 15 minute walk by walking around the circular path that begins and ends at B.
    This means that Rasheeqa’s total time, in minutes, can be each of 15,18,21,24,,96,99,102.
    Examining these lists, we see that t can equal 5,8,10,11,13,14,15, and every integer t with 16t100.
    Of the positive integers t with 1t100, we see that t cannot be equal to 1,2,3,4,6,7,9,12.
    Since there are 8 values that t cannot equal, there are 1008=92 possible values of t.

    Answer: 92

  25. The patterns that Erin can construct can include 3, 4, 5, or 6 X’s.
    A pattern cannot include fewer than 3 X’s (because 3 X’s are required to complete a pattern), and cannot include 7 X’s (because the 7th X would be placed next to 6 X’s, between 5 X’s and 1 X, or between 4 X’s and 2 X’s, or between 3 X’s and 3 X’s, each of which would already be a complete pattern).
    We consider the following cases which are determined by the number of X’s in each pattern.

    Case 1: 3 X’s

    Using O’s to represent empty squares, there are 5 ways in which 3 consecutive X’s can be placed:

    XXXOOOO, OXXXOOO, OOXXXOO, OOOXXXO, OOOOXXX

    Case 2: 4 X’s

    There are 4 ways in which 4 consecutive X’s can be placed:

    XXXXOOO, OXXXXOO, OOXXXXO, OOOXXXX

    (It is possible to have 4 consecutive X’s without having stopped after the 3rd X if the X’s are placed, for example, in order 1st X, 2nd X, 4th X, 3rd X.)
    We can also place 4 X’s with a group of 3 X’s (we need at least 3 together) and 1 separate X (either after or before). There are 12 ways in which this can be done:

    XXXOXOO, XXXOOXO, XXXOOOX, OXXXOXO, OXXXOOX, OOXXXOX, XOXXXOO, XOOXXXO, XOOOXXX, OXOXXXO, OXOOXXX, OOXOXXX

    Case 3: 5 X’s

    There are 3 ways in which 5 consecutive X’s can be placed:

    XXXXXOO, OXXXXXO, OOXXXXX

    (We can place 5 consecutive X’s in the order 1st, 2nd, 4th, 5th, 3rd.)
    We can also place 5 X’s with a group of 3 X’s and a group of 2 X’s, or with a group of 4 X’s and one individual X, or with a group of 3 X’s and two individual X’s. There are 15 ways in which this can be done:

    XXXOXXO, XXXOOXX, OXXXOXX, XXOXXXO, XXOOXXX, OXXOXXX, XXXXOXO, XXXXOOX, OXXXXOX, XOXXXXO, XOOXXXX, OXOXXXX, XXXOXOX, XOXXXOX, XOXOXXX

    Case 4: 6 X’s

    We cannot place 6 consecutive X’s, since there would have to have been 3 consecutive X’s before the 6th X was placed.
    We can, however, have 6 X’s if they are in groups of 5 and 1, or groups of 4 and 2. (We cannot have groups of 3 and 3. Can you see why?)
    There are 4 ways in which this can be done:

    XXXXXOX, XXXXOXX, XOXXXXX, XXOXXXX

    In total, there are 5+4+12+3+15+4=43 patterns that can be created.

    Answer: 43