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2021 Cayley Contest
Solutions
(Grade 10)

Thursday, February 23, 2021
(in North America and South America)

Wednesday, February 24, 2021
(outside of North American and South America)

©2020 University of Waterloo


  1. Evaluating, 2+41+2=63=2.

    Answer: (C)

  2. Since 542×3=1626, the ones digit of the result is 6.
    Note that the ones digit of a product only depends on the ones digits of the numbers being multiplied, so we could in fact multiply 2×3 and look at the ones digit of this product.

    Answer: (E)

  3. The top, left and bottom unit squares each contribute 3 sides of length 1 to the perimeter.
    The remaining square contributes 1 side of length 1 to the perimeter.
    Therefore, the perimeter is 3×3+1×1=10.
    Alternatively, the perimeter includes 3 vertical right sides, 3 vertical left sides, 2 horizontal top sides, and 2 horizontal bottom sides, which means that the perimeter is 3+3+2+2=10.

    Answer: (A)

  4. If 3x+4=x+2, then 3xx=24 and so 2x=2, which gives x=1.

    Answer: (D)

  5. Solution 1:

    10% of 500 is 110 or 0.1 of 500, which equals 50.
    100% of 500 is 500.
    Thus, 110% of 500 equals 500+50, which equals 550.

    Solution 2:

    110% of 500 is equal to 110100×500=110×5=550.

    Answer: (E)

  6. Since Eugene swam three times and had an average swim time of 34 minutes, he swam for 3×34=102 minutes in total.
    Since he swam for 30 minutes and 45 minutes on Monday and Tuesday, then on Sunday, he swam for 1023045=27 minutes.

    Answer: (C)

  7. If x=1, then x2=1 and x3=1 and so x3=x2.
    If x>1, then x3 equals x times x2; since x>1, then x times x2 is greater than x2 and so x3>x2.
    Therefore, if x is positive with x3<x2, we must have 0<x<1. We note that if 0<x<1, then both x, x2 and x3 are all positive, and x3=x2×x<x2×1=x2.
    Of the given choices, only x=34 satisfies 0<x<1, and so the answer is (B).
    We can verify by direct calculation that this is the only correct answer from the given choices.

    Answer: (B)

  8. We draw an unshaded dot to represent the location of the dot when it is on the other side of the sheet of paper being shown. Therefore, the dot moves as follows:

    First, a shaded dot is in the top right corner; second, the sheet of paper is folded along the diagonal, and an unshaded dot is in the bottom left corner; third, the folded sheet is rotated 90 degrees clockwise and the unshaded dot is in the top left corner; and fourth, the sheet of paper unfolded with a shaded dot is in the bottom right corner.

    Answer: (E)

  9. Suppose that Janice is x years old now.
    Two years ago, Janice was x2 years old.
    In 12 years, Janice will be x+12 years old.
    From the given information x+12=8(x2) and so x+12=8x16 which gives 7x=28 and so x=4.
    Checking, if Janice is 4 years old now, then 2 years ago, she was 2 years old and in 12 years she will be 16 years old; since 16=2×8, this is correct.

    Answer: (A)

  10. Join S to T and R to T.

    Since PQRS is a square, SPQ=90°.
    Since PTQ is equilateral, TPQ=60°.
    Therefore, SPT=SPQ+TPQ=90°+60°.
    Since PQRS is a square, SP=PQ.
    Since PTQ is equilateral, TP=PQ.
    Since SP=PQ and TP=PQ, then SP=TP which means that SPT is isosceles.
    Thus, PTS=12(180°SPT)=12(180°150°)=15°.
    Using a similar argument, we can show that QTR=15°.
    This means that STR=PTQPTSQTR=60°15°15°=30°.

    Answer: (D)

  11. Solution 1:

    The points A, B, C, and E can each be reached from point P by moving 3 units in either the x- or y-direction and 1 unit in the other direction.
    This means that the distance from P to each of these points is 32+12=10, using the Pythagorean Theorem.

    Therefore, the distance from P to D must be the distance that is different.

    Solution 2:

    The coordinates of the 6 points are A(2,3), B(4,5), C(6,5), D(7,4), E(8,1), P(5,2).
    Therefore, the distances from P to each of the five other points are PA=(52)2+(23)2=32+(1)2=10PB=(54)2+(25)2=12+(3)2=10PC=(56)2+(25)2=(1)2+(3)2=10PD=(57)2+(24)2=(2)2+(2)2=8PE=(58)2+(21)2=(3)2+12=10 This tells us that the distance from P to D is the one distance that is different.

    Answer: (D)

  12. Since x=2 and y=x25, then y=225=45=1.
    Since y=1 and z=y25, then z=(1)25=15=4.

    Answer: (E)

  13. Since PQR forms a straight angle, then x°+y°+x°+y°+x°=180° which gives 3x+2y=180.
    Since x+y=76 and 2x+2y+x=180, then 2(76)+x=180 or x=180152=28.

    Answer: (A)

  14. Solution 1:

    To find the x-intercept of the original line, we set y=0 to obtain 0=2x6 or 2x=6 and so x=3.
    When the line is reflected in the y-axis, its x-intercept is reflected in the y-axis to become x=3.

    Solution 2:

    When a line is reflected in the y-axis, its slope changes signs (that is, is multiplied by 1).
    When a line is reflected in the y-axis, its y-intercept (which is on the y-axis) does not change.
    Thus, when the line with equation y=2x6 is reflected in the y-axis, the equation of the resulting line is y=2x6.
    To find the x-intercept of this line, we set y=0 to obtain 0=2x6 or 2x=6 and so x=3.

    Answer: (D)

  15. Amy bought and then sold 15n avocados.
    Since she bought the avocados in groups of 3, she bought 15n3=5n groups of 3 avocados.
    Since she paid $2 for every 3 avocados, she paid $2×5n=$10n.
    Since she sold the avocados in groups of 5, she sold 15n5=3n groups of 5 avocados.
    Since she sold every 5 avocados for $4, she received $4×3n=$12n.
    In terms of n, Amy’s profit is $12n$10n=$2n.
    Since we know that Amy’s profit was $100, we get $2n=$100 and so 2n=100 or n=50.

    Answer: (C)

  16. Using exponent laws, 3x+2=3x32=3x9.
    Since 3x=5, then 3x+2=3x9=59=45.

    Answer: (E)

  17. We work backwards through the given information.
    At the end, there is 1 candy remaining.
    Since 56 of the candies are removed on the fifth day, this 1 candy represents 16 of the candies left at the end of the fourth day.
    Thus, there were 6×1=6 candies left at the end of the fourth day.
    Since 45 of the candies are removed on the fourth day, these 6 candies represent 15 of the candies left at the end of the third day.
    Thus, there were 5×6=30 candies left at the end of the third day.
    Since 34 of the candies are removed on the third day, these 30 candies represent 14 of the candies left at the end of the second day.
    Thus, there were 4×30=120 candies left at the end of the second day.
    Since 23 of the candies are removed on the second day, these 120 candies represent 13 of the candies left at the end of the first day.
    Thus, there were 3×120=360 candies left at the end of the first day.
    Since 12 of the candies are removed on the first day, these 360 candies represent 12 of the candies initially in the bag.
    Thus, there were 2×360=720 in the bag at the beginning.

    Answer: (B)

  18. Elina and Gustavo start by running and walking for 12 minutes.
    Since there are 60 minutes in 1 hour, 12 minutes equals 15 of an hour.
    When Elina runs at 12 km/h for 15 of an hour, she runs 12 km/h×15 h=2.4 km to the north.
    When Gustavo walks at 5 km/h for 15 of an hour, he walks 5 km/h×15 h=1 km to the east.
    At this point, Elina and Gustavo start to travel directly towards each other.

    The legs of a right-angled triangle show Elina and Gustavo's initial paths from the school respectively. The leg pointing north is labelled 2.4 km and has E as its top endpoint. The leg pointing is labelled 1 km and has G as its rightmost endpoint. The hypotenuse joins points E and G with a dotted line.

    As they change direction, we use the Pythagorean Theorem to calculate their distance from each other. Using this, we obtain (2.4 km)2+(1 km)2=2.6 km.
    Since Elina continues to travel at 12 km/h, Gustavo continues to travel at 5 km/h, and they travel directly towards each other, they close the gap at a rate of 12 km/h+5 km/h=17 km/h.
    Thus, it takes 2.6 km17 km/h0.153 h for them to meet.
    Since there are 60 minutes in an hour, 0.153 h is equivalent to roughly 9.18 minutes.
    Since Elina and Gustavo leave at 3:00 p.m. and travel for 12 minutes and then for an additional 9 minutes, they meet again at approximately 3:21 p.m.

    Answer: (E)

  19. Each of the four shaded circles has radius 1, and so has area π12 which equals π.
    Next, we consider one of the three spaces. By symmetry, each of the three spaces has the same area.
    Consider the leftmost of the three spaces.
    Join the centres of the circles that bound this space to form a quadrilateral. Also, join the centres of these circles to the points where these circles touch the sides of the rectangle.

    A description of the diagram follows.

    This quadrilateral is a square with side length 2.
    The side length of this quadrilateral is 2, because each of MD, DE, EN, and NM is equal to the sum of lengths of two radii, or 2.
    Next, we show that the angles of DMNE are each 90°.
    Consider quadrilateral ABML. The angle at A is 90°, since the larger shape is a rectangle. The angles at B and L are both 90° since radii are perpendicular to tangents at their points of tangency. Thus, ABML has three 90° angles, which means that its fourth angle must also be 90°.
    Now consider quadrilateral BCDM. The angles at B and C are both 90°, as above. Since BM=CD=1, this means that BCMD is actually a rectangle.
    In a similar way, we can see that JNGH is a square and MLJN is a rectangle.
    Finally, we can consider DMNE. At M, the three angles outside DMNE are each 90°, which means that the angle at M inside DMNE is 90°.
    Similarly, the angle at N inside DMNE is 90°.
    Since these angles are both 90° and MD=NE=2, then DMNE is a rectangle. Since its four sides each have length 2, then this rectangle must be a square.
    The area of the space between the four circles is thus equal to the area of the square minus the area of the four circular sectors inside the square. In fact, each of these four circular sectors is one-quarter of a circle of radius 1, since its angle at the centre of the circle is 90°.
    Thus, the area of the space is equal to 22414π12 which equals 4π.
    This means that the total area of the shaded region equals 4π+3(4π)=4π+123π=12+π which is approximately equal to 15.14.
    Of the given choices, this is closest to 15.

    Answer: (D)

  20. An integer is divisible by both 12 and 20 exactly when it is divisible by the least common multiple of 12 and 20.
    The first few positive multiples of 20 are 20, 40, 60. Since 60 is divisible by 12 and neither 20 nor 40 is divisible by 12, then 60 is the least common multiple of 12 and 20.
    Since 6016=960 and 6017=1020, the smallest four-digit multiple of 60 is 6017.
    Since 60166=9960 and 60167=10020, the largest four-digit multiple of 60 is 60166.
    This means that there are 16617+1=150 four-digit multiples of 60.
    Now, we need to remove the multiples of 60 that are also multiples of 16.
    Since the least common multiple of 60 and 16 is 240, we need to remove the four-digit multiples of 240.
    Since 2404=960 and 2405=1200, the smallest four-digit multiple of 240 is 2405.
    Since 24041=9840 and 24042=10080, the largest four-digit multiple of 240 is 24041.
    This means that there are 415+1=37 four-digit multiples of 240.
    Finally, this means that the number of four-digit integers that are multiples of 12 and 20 but are not multiples of 16 is 15037=113.

    Answer: (B)

  21. We systematically work through pairs of the given integers to see which pairs add up to a third given integer. Starting with the smallest possible pairs, we have: 4+27=3112+15=2712+27=39 There are no other pairs that add up to a third given integer.
    This means that each of the three sums in the problem must be one of the three sums above.
    In the sums above, the only integer that appears three times is 27.
    In the sums in the problem, the only variable that appears three times is c.
    Therefore, c=27.
    This also means that the sum a+b=c must be the sum 12+15=27.
    Since 12 appears in two sums and 15 does not, then a=15 and b=12.
    Matching the values that we know already with the equations that we have, we obtain a+b=c15+12=27b+c=d12+27=39c+e=f27+4=31 Therefore, a+c+f=15+27+31=73.

    Answer: (C)

  22. Suppose that the integer in the bottom left corner is n.
    In this case, the sum of the integers in the first column is 64+70+n or n+134.
    Thus, the sum of the integers in each row, in each column, and on each diagonal also equals n+134. (This means that this square is in fact a magic square, since the sum of the numbers in each row, in each column, and on each diagonal is the same.)
    Using the top row, the top right integer equals (n+134)6410 or n+60.
    Using the northeast diagonal, the centre integer equals (n+134)n(n+60) or 74n.
    Using the second row, the middle integer in the right column equals (n+134)70(74n) or 2n10.
    Using the southeast diagonal, the bottom right integer equals (n+134)64(74n) or 2n4.
    Using the third row, the middle integer is x=(n+134)n(2n4)=1382n. 6410n+607074n2n10n1382n2n4 Using the third column, (n+60)+(2n10)+(2n4)=n+1345n+46=n+1344n=88n=22 Therefore, x=1302n=13044=94 and the complete grid is 641082705234229440 .

    Answer: (E)

  23. Robbie has a score of 8 and Francine has a score of 10 after two rolls each. Thus, in order for Robbie to win (that is, to have a higher total score), his third roll must be at least 3 larger than that of Francine.
    If Robbie rolls 1, 2 or 3, his roll cannot be 3 larger than that of Francine.
    If Robbie rolls a 4 and wins, then Francine rolls a 1.
    If Robbie rolls a 5 and wins, then Francine rolls a 1 or a 2.
    If Robbie rolls a 6 and wins, then Francine rolls a 1 or a 2 or a 3.
    We now know the possible combinations of rolls that lead to Robbie winning, and so need to calculate the probabilities.
    We recall that Robbie and Francine are rolling a special six-sided dice.
    Suppose that the probability of rolling a 1 is p.
    From the given information, the probability of rolling a 2 is 2p, of rolling a 3 is 3p, and of rolling a 4, 5 and 6 is 4p, 5p and 6p, respectively.
    Since the combined probability of rolling a 1, 2, 3, 4, 5, or 6 equals 1, we get the equation p+2p+3p+4p+5p+6p=1 which gives 21p=1 or p=121.
    Thus, the probability that Robbie rolls a 4 and Francine rolls a 1 equals the product of the probabilities of each of these events, which equals 421121.
    Also, the probability that Robbie rolls a 5 and Francine rolls a 1 or 2 equals 521121+521221.
    Lastly, the probability that Robbie rolls a 6 and Francine rolls a 1, a 2, or a 3 equals 621121+621221+621321 Therefore, the probability that Robbie wins is 421121+521121+521221+621121+621221+621321=4+5+10+6+12+182121=55441 which is in lowest terms since 55=511 and 441=3272.
    Converting to the desired form, we see that r=55 and s=41 which gives r+s=96.

    Answer: (A)

  24. Let O be the centre of the top face of the cylinder and let r be the radius of the cylinder.
    We need to determine the value of QT2.
    Since RS is directly above PQ, then RP is perpendicular to PQ.
    This means that TPQ is right-angled at P.
    Since PQ is a diameter, then PQ=2r.
    By the Pythagorean Theorem, QT2=PT2+PQ2=n2+(2r)2=n2+4r2.
    So we need to determine the values of n and r. We will use the information about QU and UT to determine these values.
    Join U to O.
    Since U is halfway between R and S, then the arcs RU and US are each one-quarter of the circle that bounds the top face of the cylinder.
    This means that UOR=UOS=90°.

    We can use the Pythagorean Theorem in UOR and UOS, which are both right-angled at O, to obtain UR2=UO2+OR2=r2+r2=2r2andUS2=2r2 Since RP and QS are both perpendicular to the top face of the cylinder, we can use the Pythagorean Theorem in TRU and in QSU to obtain QU2=QS2+US2=m2+2r2UT2=TR2+UR2=(PRPT)2+2r2=(QSn)2+2r2=(mn)2+2r2 Since QU=933, then QU2=9233=2673.
    Since UT=40, then UT2=1600.
    Therefore, m2+2r2=2673(mn)2+2r2=1600 Subtracting the second equation from the first, we obtain the equivalent equations m2(mn)2=1073m2(m22mn+n2)=10732mnn2=2937n(2mn)=2937 Since m and n are integers, then 2mn is an integer. Thus, n and 2mn are a factor pair of 2937=1073.
    Since 29 and 37 are prime numbers, the integer 1073 has only four positive divisors: 1, 29, 37, 1073.
    This gives the following possibilities:

    n 2mn m=((2mn)+n)
    1 1073 537
    29 37 33
    37 29 33
    1073 1 537

    Since m>n, then n cannot be 37 or 1073.
    Since QU>QS, then m<93351.7.
    This means that n=29 and m=33.
    Since (mn)2+2r2=1600, we obtain 2r2=1600(mn)2=160042=1584 and so QT2=n2+4r2=292+2(2r2)=841+3168=4009 The remainder when QT2 is divided by 100 is 9.

    Answer: (C)

  25. The distance between J(2,7) and K(5,3) is equal to (25)2+(73)2=32+42=5.
    Therefore, if we consider JKL as having base JK and height h, then we want 12JKh10 which means that h1025=4.
    In other words, L(r,t) can be any point with 0r10 and 0t10 whose perpendicular distance to the line through J and K is at most 4.

    Points J, K, and L are plotted on the first quadrant of the xy-plane. A line is drawn through J and K and extended so that it touches each of the axes. Point L is not on the line, and triangle JKL is drawn with height h, from L to the line.

    The slope of the line through J(2,7) and K(5,3) is equal to 7325=43.
    Therefore, this line has equation y7=43(x2).
    Muliplying through by 3, we obtain 3y21=4x+8 or 4x+3y=29.
    We determine the equation of the line above this line that is parallel to it and a perpendicular distance of 4 from it.
    The equation of this line will be of the form 4x+3y=c for some real number c, since it is parallel to the line with equation 4x+3y=29.
    To determine the value of c, we determine the coordinates of one point on this line.
    To determine such a point, we draw a perpendicular of length 4 from K to a point P above the line.
    Since JK has slope 43 and KP is perpendicular to JK, then KP has slope 34.
    Draw a vertical line from P and a horizontal line from K, meeting at Q.

    Triangle KPQ is a right-angled triangle with the right angle at Q.

    Since KP has slope 34, then PQ:QK=3:4, which means that KQP is similar to a 3-4-5 triangle.
    Since KP=4, then PQ=35KP=125 and QK=43KP=165.
    Thus, the coordinates of P are (5+165,3+125) or (415,275).
    Since P lies on the line with equation 4x+3y=c, then c=4415+3275=1645+815=2455=49 and so the equation of the line parallel to JK and 4 units above it is 4x+3y=49.
    In a similar way, we find that the line parallel to JK and 4 units below it has equation 4x+3y=9. (Note that 4929=299.)
    This gives us the following diagram:

    Two lines are drawn parallel to JK. The first line passes through the points (19/4, 10) and (10,3). The second line passes through the points (0,3) and (9/4, 0).

    The points L that satisfy the given conditions are exactly the points within the square, below the line 4x+3y=49 and above the line 4x+3y=9. In other words, the region R is the region inside the square and between these lines.
    To find the area of R, we take the area of the square bounded by the lines x=0, x=10, y=0, and y=10 (this area equals 1010 or 100) and subtract the area of the two triangles inside the square and not between the lines.
    The line with equation 4x+3y=9 intersects the y-axis at (0,3) (we see this by setting x=0) and the x-axis at (94,0) (we see this by setting y=0).
    The line with equation 4x+3y=49 intersects the line x=10 at (10,3) (we see this by setting x=10) and the line y=10 at (194,10) (we see this by setting y=10).
    The bottom triangle that is inside the square and outside R has area 12394=278.
    The top triangle that is inside the square and outside R has horizontal base of length 10194 or 214 and vertical height of length 103 or 7, and thus has area 122147=1478.
    Finally, this means that the area of R is 1002781478=1001748=100874=3134 which is in lowest terms since the only divisors of the denominator that are larger than 1 are 2 and 4, while the numerator is odd.
    When we write this area in the form 300+a40b where a and b are positive integers, we obtain a=13 and b=36, giving a+b=49.

    Answer: (D)