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2023 Cayley Contest
Solutions
(Grade 10)

Wednesday, February 22, 2023
(in North America and South America)

Thursday, February 23, 2023
(outside of North American and South America)

©2022 University of Waterloo


  1. Since each of the three fractions is equal to 1, then 11+22+33=1+1+1=3.

    Answer: (C)

  2. Since 3n=9+9+9=3×9, then n=9.
    Alternatively, we could note that 9+9+9=27 and so 3n=27 which gives n=273=9.

    Answer: (D)

  3. We add 25 minutes to 1 hour and 48 minutes in two steps.
    First, we add 12 minutes to 1 hour and 48 minutes to get 2 hours.
    Then we add 2512=13 minutes to 2 hours to get 2 hours and 13 minutes.
    Alternatively, we could note that 1 hour and 48 minutes is 60+48=108 minutes, and so the time that is 25 minutes longer is 133 minutes, which is 120+13 minutes or 2 hours and 13 minutes.

    Answer: (A)

  4. On Day 1, Lucy sees 2 blue jays and 3 cardinals, and so sees 1 more cardinal than blue jay.
    On Day 2, Lucy sees 3 blue jays and 3 cardinals.
    On Day 3, Lucy sees 2 blue jays and 4 cardinals, and so sees 2 more cardinals than blue jays.
    Thus, over the three days, Lucy saw 1+0+2=3 more cardinals than blue jays.

    Answer: (B)

  5. When looking at From left to right, 2, 0, 2, 3. through a window from behind, the digits appear as mirror images and in reverse order, and so appear as From left to right, the mirror image of 3, the mirror image of 2, 0, the mirror image of 2..

    Answer: (C)

  6. Solution 1

    Since BCD is a straight angle, then ACB=180°ACD=180°150°=30°.
    Since the measures of the angles in ABC add to 180°, then x°+90°+30°=180° and so x=180120=60.

    Solution 2

    Since the measures of the angles in ABC add to 180°, then ACB=180°ABCBAC=180°90°x°=90°x° Since BCD=180°, then ACB+ACD=180°(90°x°)+150°=180°240x=180 and so x=60.

    Answer: (E)

  7. A cube has six identical faces.
    If the surface area of a cube is 24, the area of each face is 246=4.
    Since each face of this cube is a square with area 4, the edge length of the cube is 4=2.
    Thus, the volume of the cube is 23 which equals 8.

    Answer: (E)

  8. If 45 of the beads are yellow, then 15 are green.
    Since there are 4 green beads, the total number of beads must be 4×5=20.
    Thus, Charlie needs to add 204=16 yellow beads.

    Answer: (A)

  9. Solution 1

    Suppose that the original number is 100.
    When 100 is increased by 60%, the result is 160.
    To return to the original value of 100, 160 must be decreased by 60.
    This percentage is 60160×100%=38×100%=37.5%.

    Solution 2

    Suppose that the original number is x for some x>0.
    When x is increased by 60%, the result is 1.6x.
    To return to the original value of x, 1.6x must be decreased by 0.6x.
    This percentage is 0.6x1.6x×100%=38×100%=37.5%.

    Answer: (E)

  10. Since each door can be open or closed, there are 2 possible states for each door.
    Since there are 5 doors, there are 25=32 combinations of states for the 5 doors.
    If the doors are labelled P, Q, R, S, T, the pairs of doors that can be opened are PQ, PR, PS, PT, QR, QS, QT, RS, RT, ST. There are 10 such pairs.
    Therefore, if one of the 32 combinations of states is chosen at random, the probability that exactly two doors are open is 1032 which is equivalent to 516.

    Answer: (A)

  11. After Karim eats n candies, he has 23n candies remaining.
    Since he divides these candies equally among his three children, the integer 23n must be a multiple of 3.
    If n=2,5,11,14, we obtain 23n=21,18,12,9, each of which is a multiple of 3.
    If n=9, we obtain 23n=14, which is not a multiple of 3.
    Therefore, n cannot equal 9.

    Answer: (C)

  12. A rectangle that is 6 m by 8 m has perimeter 2×(6 m+8 m)=28 m.
    If posts are put in every 2 m around the perimeter starting at a corner, then we would guess that it will take 28 m2 m=14 posts.
    The diagram below confirms this:

    A rectangle with 14 points equally spaced around the
perimeter. There is 1 point at each vertex, 3 additional points on each
of the top and bottom side, and 2 points on each of the left and right
side.

    Answer: (B)

  13. Since 2023=7×172, then any perfect square that is a multiple of 2023 must have prime factors of both 7 and 17.
    Furthermore, the exponents of the prime factors of a perfect square must be all even.
    Therefore, any perfect square that is a multiple of 2023 must be divisible by 72 and by 172, and so it is at least 72×172 which equals 7×2023.
    Therefore, the smallest perfect square that is a multiple of 2023 is 7×2023.
    We can check that 20232 is larger than 7×2023 and that none of 4×2023 and 17×2023 and 7×17×2023 is a perfect square.

    Answer: (C)

  14. Since B is between A and D and BD=3AB, then B splits AD in the ratio 1:3.
    Since AD=24, then AB=6 and BD=18.
    Since C is halfway between B and D, then BC=12BD=9.

    Thus, AC=AB+BC=6+9=15.

    Answer: (E)

  15. Since a=1n where n is a positive integer with n>1, then 0<a<1 and 1a=n>1.
    Thus, 0<a<1<1a, which eliminates choices (D) and (E).
    Since 0<a<1, then a2 is positive and a2<a, which eliminates choices (A) and (C).
    Thus, 0<a2<a<1<1a, which tells us that (B) must be correct.

    Answer: (B)

  16. Solution 1

    Since AB and ED are parallel, quadrilateral ABDE is a trapezoid.
    We know that AB=30 cm.
    Since ABCF is a rectangle, then FC=AB=30 cm.
    Suppose that DC=x cm.
    Then ED=FCFEDC=(30 cm)(5 cm)(x cm)=(25x) cm.
    The height of the trapezoid is the length of AF, which is 14 cm.
    Since the area of the trapezoid is 266 cm2, then 266 cm2=30 cm+(25x) cm2×(14 cm)266=55x2×14532=(55x)×1438=55xx=5538 and so DE=x cm=17 cm.

    Solution 2

    Let DC=x cm.
    Rectangle ABCF has AB=30 cm and AF=14 cm, and so the area of ABCF is (30 cm)×(14 cm)=420 cm2.
    The area of AFE, which is right-angled at F, is 12×AF×FE=12×(14 cm)×(5 cm)=35 cm2 The area of quadrilateral ABDE is 266 cm2.
    The area of BCD, which is right-angled at C, is 12×BC×DC=12×(14 cm)×(x cm)=7x cm2 Comparing the area of rectangle ABCF to the combined areas of the pieces, we obtain (35 cm2)+(266 cm2)+(7x cm2)=420 cm2301+7x=4207x=119x=17 Thus, the length of DC is 17 cm.

    Answer: (A)

  17. Megan’s car travels 100 m at 54 m/s, and so takes 100 m5/4 m/s=4005 s=80 s.

    Hana’s car completes the 100 m in 5 s fewer, and so takes 75 s.

    Thus, the average speed of Hana’s car was 100 m75 s=10075 m/s=43 m/s.

    Answer: (C)

  18. The number of bars taken from the boxes is 1+2+4+8+16=31.
    If these bars all had mass 100 g, their total mass would be 3100 g.
    Since their total mass is 2920 g, they are 3100 g2920 g=180 g lighter.
    Since all of the bars have a mass of 100 g or of 90 g, then it must be the case that 18 of the bars are each 10 g lighter (that is, have a mass of 90 g).
    Thus, we want to write 18 as the sum of two of 1, 2, 4, 8, 16 in order to determine the boxes from which the 90 g bars were taken.
    We note that 18=2+16 and so the 90 g bars must have been taken from box W and box Z. Can you see why this is unique?

    Answer: (B)

  19. Since the average of a, b and c is 16, then a+b+c3=16 and so a+b+c=3×16=48.

    Since the average of c, d and e is 26, then c+d+e3=26 and so c+d+e=3×26=78.

    Since the average of a, b, c, d, and e is 20, then a+b+c+d+e5=20.

    Thus, a+b+c+d+e=5×20=100.
    We note that (a+b+c)+(c+d+e)=(a+b+c+d+e)+c and so 48+78=100+c which gives c=126100=26.

    Answer: (D)

  20. Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south.
    In other words, we can consider each group of four jumps, starting with the first, as resulting in a net movement of 2 cm to the west and 2 cm to the south.
    We note that 158=2×79.
    Thus, after 79 groups of four jumps, the grasshopper is 79×2=158 cm to the west and 158 cm to the south of its original position. (We need at least 79 groups of these because the grasshopper cannot be 158 cm to the south of its original position before the end of 79 such groups.)
    The grasshopper has made 4×79=316 jumps so far.
    After the 317th jump (1 cm to the east), the grasshopper is 157 cm west and 158 cm south of its original position.
    After the 318th jump (2 cm to the north), the grasshopper is 157 cm west and 156 cm south of its original position.
    After the 319th jump (3 cm to the west), the grasshopper is 160 cm west and 156 cm south of its original position.
    After the 320th jump (4 cm to the south), the grasshopper is 160 cm west and 160 cm south of its original position.
    After the 321st jump (1 cm to the east), the grasshopper is 159 cm west and 160 cm south of its original position.
    After the 322nd jump (2 cm to the north), the grasshopper is 159 cm west and 158 cm south of its original position.
    After the 323rd jump (3 cm to the west), the grasshopper is 162 cm west and 158 cm south of its original position, which is the desired position.
    As the grasshopper continues jumping, each of its positions will always be at least 160 cm south of its original position, so this is the only time that it is at this position.
    Therefore, n=323. The sum of the squares of the digits of n is 32+22+32=9+4+9=22.

    Answer: (A)

  21. Since the line with equation y=mx50 passes through the point (a,0), then 0=ma50 or ma=50.
    Since m and a are positive integers whose product is 50, then m and a are a divisor pair of 50.
    Therefore, the possible values of m are the positive divisors of 50, which are 1, 2, 5, 10, 25, 50.
    The sum of the possible values of m is thus 1+2+5+10+25+50=93.

    Answer: 93

  22. From the given information, if a and b are in two consecutive squares, then a+b goes in the circle between them.
    Since all of the numbers that we can use are positive, then a+b is larger than both a and b.
    This means that the largest integer in the list, which is 13, cannot be either x or y (and in fact cannot be placed in any square). This is because the number in the circle next to it must be smaller than 13 (because 13 is the largest number in the list) and so cannot be the sum of 13 and another positive number from the list.
    Thus, for x+y to be as large as possible, we would have x and y equal to 10 and 11 in some order. But here we have the same problem: there is only one larger number from the list (namely 13) that can go in the circles next to 10 and 11, and so we could not fill in the circle next to both 10 and 11.
    Therefore, the next largest possible value for x+y is when x=9 and y=11. (We could also swap x and y.)
    Here, we could have 13=11+2 and 10=9+1, giving the following partial list:

    From the left to right, the alternating squares and circles contain: 11, 13, 2, empty, empty, empty, 1, 10, 9.

    The remaining integers (4, 5 and 6) can be put in the shapes in the following way that satisfies the requirements.

    From left to right, the alternative squares and circles contain: 11, 13, 2, 6, 4, 5, 1, 10, 9.

    This tells us that the largest possible value of x+y is 20.

    Answer: 20

  23. Since AB and BC are both radii of the circle, then AB=BC.
    Since ABC is a quarter-circle centred at B, then ABC=90°.
    Thus, ABC is isosceles and right-angled, which means that BAC=BCA=45°.
    We add some additional labels to the diagram:

    D is the intersection of AC and PQ. E is the intersection of AC and QR. F is the intersection of AC and RU. G is the intersection of AC and UV.

    We note that the angles between the straight lines at P, Q, R, and U are all right angles.
    Since PAD=BAC=45°, this means that PAD=ADP=QDE=DEQ=REF=EFR=UFG=UGF=45° This in turn means that each of APD, DQE, ERF, and FUG is right-angled and isosceles.
    Since the side length of each square is 10, then BT=10 and TQ=TR+RQ=20.
    Since BTQ=90°, then by the Pythagorean Theorem, BQ=BT2+TQ2=102+202=500 We note that 500=100×5=100×5=105.
    Since BQ is a radius of the circle, then BQ=BA=BC=105.
    Since BP=TQ=20, then AP=BABP=10520.
    Thus, PD=AP=10520.
    Since PQ=10, then DQ=PQPD=10(10520)=30105.
    Thus, QE=DQ=30105.
    Since PQ=QR and DQ=QE, then PD=ER=10520.
    Using similar reasoning, ER=RF=10520 and UF=UG=30105.
    The total area, A, of the shaded regions equals the sum of the areas of DQE, ERF and FUG.
    Therefore, A=12×DQ×QE+12×ER×RF+12UF×UG=12(30105)2+12(10520)2+12(30105)2=(30105)2+12(10520)2=(3022×30×105+(105)2)+12((105)22×105×20+202)=(9006005+500)+12(5004005+400)=14006005+4502005=1850800561.14 and so the integer closest to A is 61.

    Answer: 61

  24. We want to determine the probability that Carina wins 3 games before she loses 2 games.
    This means that she either wins 3 and loses 0, or wins 3 and loses 1.
    If Carina wins her first three games, we do not need to consider the case of Carina losing her fourth game, because we can stop after she wins 3 games.
    Putting this another way, once Carina has won her third game, the outcomes of any later games do not affect the probability because wins or losses at that stage will not affect the question that is being asked.
    Using W to represent a win and L to represent a loss, the possible sequence of wins and losses that we need to examine are WWW, LWWW, WLWW, and WWLW.

    In the case of WWW, the probabilities of the specific outcome in each of the three games are 12, 34, 34, because the probability of a win after a win is 34.

    Therefore, the probability of WWW is 12×34×34=932.

    In the case of LWWW, the probabilities of the specific outcome in each of the four games are 12, 13, 34, 34, because the probability of a loss in the first game is 12, the probability of a win after a loss is 13, and the probability of a win after a win is 34.

    Therefore, the probability of LWWW is 12×13×34×34=996=332.

    Using similar arguments, the probability of WLWW is 12×14×13×34=396=132.

    Here, we used the fact that the probability of a loss after a win is 134=14.

    Finally, the probability of WWLW is 12×34×14×13=396=132.

    Therefore, the probability that Carina wins 3 games before she loses 2 games is 932+332+132+132=1432=716, which is in lowest terms.
    The sum of the numerator and denominator of this fraction is 23.

    Answer: 23

  25. Solution 1

    Let N=AB0AB and let t be the two-digit integer AB.
    We note that N=1001t, and that 1001=1191=11713.
    Therefore, N=t71113.
    We want to write N as the product of 5 distinct odd integers, each greater than 2, and to count the number of sets S of such odd integers whose product is N.
    There are several situations to consider.
    First, we look at possible sets S that include the integers 7, 11 and 13 (which we know are divisors).
    Second, we look at possible sets S that include two of these integers and an odd multiple of the third.
    Third, we rule out possible sets S that include one of these integers and odd multiples of the second and third.
    Fourth, we rule out possible sets S that include the product of two or three of these integers and additional integers.

    Case 1: S={7,11,13,m,n} where m<n and m,n7,11,13.

    Here, N=71113mn=mn1001 and so t=mn. This tells us that mn is less than 100.
    If m=3, then the possible values for n are 5, 9, 15, 17, 19, 21, 23, 25, 27, 29, 31.
    These give the following corresponding values of mn: 15, 27, 45, 51, 57, 63, 69, 75, 81, 87, 93.
    Note that n33, since m=3 and n=33 gives mn=99 which has two equal digits and so is not possible.
    If m=5, then the possible values for n are 9, 15, 17, 19.
    If m9, then n15 since the integers in n are odd and distinct, and so mn135, which is not possible.
    Therefore, in this case, there are 15 possible sets.

    Case 2: S={7q,11,13,m,n} where m<n and q>1 is odd and m,n7q,11,13.

    Here, we have N=7q1113mn and so N=1001mnq which gives t=mnq.
    Note that mnq99.
    Suppose that q=3. This means that mn33.
    If m=3, then the possible values of n are 5 and 9 since m and n are odd, greater than 2, and distinct. (n=7 is not possible, since this would give the set {21,11,13,3,7} which is already counted in Case 1 above.)
    If m5, then n7 which gives mn35, which is not possible.
    Suppose that q=5. This means that mn995=1945.
    If m=3, then n=5. There are no further possibilities when q=5.
    Since mn35=15 and mnq99, then we cannot have q7.
    Therefore, in this case, there are 3 possible sets.

    Case 3: S={7,11q,13,m,n} where m<n and q>1 is odd and m,n7,11q,13.

    Suppose that q=3. This means that mn33.
    If m=3, then the possible values of n are 5 and 9. (Note that n7.) We cannot have n=11 as this would give mnq=99 and a product of 99099 which has equal digits A and B.
    We cannot have m5 since this gives mn45.
    Suppose that q=5. This means that mn995.
    If m=3, then n=5.
    As in Case 2, we cannot have q7.
    Therefore, in this case, there are 3 possible sets.

    Case 4: S={7,11,13q,m,n} where m<n and q>1 is odd and m,n7,11,13q.

    Suppose that q=3. This means that mn33.
    If m=3, the possible values of n are 5 and 9. (Again, n11 in this case.)
    We cannot have m5 when q=3 otherwise mn45.
    If q=5, we can have m=3 and n=5 but there are no other possibilities.
    As in Cases 2 and 3, we cannot have q7.
    Therefore, in this case, there are 3 possible sets.

    Case 5: S={7q,11r,13,m,n} where m<n and q,r>1 are odd and m,n7q,11r,13.

    Here, mnqr99.
    Since q,r>1 are odd, then qr9 which means that mn11.
    Since there do not exist two distinct odd integers greater than 1 with a product less than 15, there are no possible sets in this case.
    A similarly argument rules out the products N=7q1113rmnN=711q13rmnN=7q11r13smn where q, r, s are odd integers greater than 1.

    Case 6: S={77,13,m,n,} where m<n< and m,n,77,13.

    Note that 77=711 since we know that N has divisors of 7 and 11.
    Here, mn99.
    Since mn357=105, there are no possible sets in this case, nor using 7143 or 1191 in the product or 1001 by itself or multiples of 77, 91 or 143.

    Having considered all cases, there are 15+3+3+3=24 possible sets.

    Solution 2

    We note first that AB0AB=AB1001, and that 1001=1191=11713.
    Therefore, AB0AB=AB71113.
    Since AB0AB is odd, then B is odd.
    Since A0 and AB and B is odd, then we have the following possibilities for the two-digit integer AB: 13,15,17,19,21,23,25,27,29,31,35,37,39,41,43,45,47,49,51,53,57,59,61,63,65,67,69,71,73,75,79,81,83,85,87,89,91,93,95,97 If the integer AB is a prime number, then AB0AB cannot be written as the product of five different positive integers each greater than 2, since it would have at most four prime factors.
    Using this information, we can eliminate many possibilities for AB from our list to obtain the shorter list: 15,21,25,27,35,39,45,49,51,57,63,65,69,75,81,85,87,91,93,95 Several of the integers in this shorter list are the product of two distinct prime numbers neither of which is equal to 7, 11 or 13. These integers are 15=35 and 51=317 and 57=319 and 69=323 and 85=517 and 87=329 and 93=331 and 95=519.
    Thinking about each of these as pq for some distinct prime numbers p and q, we have AB0AB=pq71113.
    To write AB0AB as the product of five different positive odd integers greater each greater than 2, these five integers must be the five prime factors. For each of these 8 integers (15, 51, 57, 69, 85, 87, 93, 95), there is 1 set of five distinct odd integers, since the order of the integers does not matter. This is 8 sets so far.
    This leaves the integers 21, 25, 27, 35, 39, 45, 49, 63, 65, 75, 81, 91.

    Seven of these remaining integers are equal to the product of two prime numbers, which are either equal primes or at least one of which is equal to 7, 11 or 13. These products are 21=37 and 25=55 and 35=57 and 39=313 and 49=77 and 65=513 and 91=713.
    In each case, AB0AB can then be written as a product of 5 prime numbers, at least 2 of which are the same. These 5 prime numbers cannot be grouped to obtain five different odd integers, each larger than 1, since the 5 prime numbers include duplicates and if two of the primes are combined, we must include 1 in the set. Consider, for example, 21=37. Here, 21021=3771113. There is no way to group these prime factors to obtain five different odd integers, each larger than 1.
    Similarly, 25025=5571113 and 91091=71371113. The remaining three possibilities (35, 49 and 65) give similar situations.
    This leaves the integers 27, 45, 63, 75, 81 to consider.

    Consider 27027=3371113. There are 6 prime factors to distribute among the five odd integers that form the product. Since there cannot be two 3’s in the set, the only way to do this so that they are all different is {3,9,7,11,13}.
    Consider 81081=3471113. There are 7 prime factors to distribute among the five odd integers that form the product.
    Since there cannot be two 3s or two 9s in the set and there must be two powers of 3 in the set, there are four possibilities for the set S: S={3,27,7,11,13},{3,9,21,11,13},{3,9,7,33,13},{3,9,7,11,39} Consider 45045=32571113. There are 6 prime factors to distribute among the five odd integers that form the product.
    Since two of these prime factors are 3, they cannot each be an individual element of the set and so one of the 3s must always be combined with another prime giving the following possibilities: S={9,5,7,11,13},{3,15,7,11,13},{3,5,21,11,13},{3,5,7,33,13},{3,5,7,11,39} Consider 75075=35271113.
    Using a similar argument to that in the case of 45045, we obtain S={15,5,7,11,13},{3,25,7,11,13},{3,5,35,11,13},{3,5,7,55,13},{3,5,7,11,65} Finally, consider 63063=32721113. There are 6 prime factors to distribute among the five odd integers that form the product. Since we cannot have two 3s or two 7s in the product, the second 3 and the second 7 must be combined, and so there is only one set in this case, namely S={3,7,21,11,13}.

    We have determined that the total number of sets is thus 8+1+4+5+5+1=24.

    Answer: 24