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2022 Cayley Contest
Solutions
(Grade 10)

Wednesday, February 23, 2022
(in North America and South America)

Thursday, February 24, 2022
(outside of North American and South America)

©2021 University of Waterloo


  1. Evaluating, 2+(0×22)=2+0=2.

    Answer: (B)

  2. The ones digit of 119 is not even, so 119 is not a multiple of 2.
    The ones digit of 119 is not 0 or 5, so 119 is not a multiple of 5.
    Since 120=3×40, then 119 is 1 less than a multiple of 3 so is not itself a multiple of 3.
    Since 110=11×10 and 121=11×11, then 119 is between two consecutive multiples of 11, so is not itself a multiple of 11.
    Finally, 119÷7=17, so 119 is a multiple of 7.

    Answer: (D)

  3. The fractions 310 and 523 are each less than 12 (which is choice (E)) so cannot be the greatest among the choices. (Note that 12=510=11.523 which we can use to compare the given fractions to 12.)
    The fractions 47 and 23 are each greater than 12, so 12 cannot be the greatest among the choices.
    This means that the answer must be either 47 or 23.
    Using a common denominator of 3×7=21, we re-write these fractions as 47=1221 and 23=1421 which shows that 23 has the gretest value among the five choices.
    (Alternatively, we could have converted the fractions into decimals and used their decimal approximations to compare their sizes.)

    Answer: (D)

  4. The sequence consists of a pattern of 5 shapes that are repeated.
    The first repetitions of this pattern end on the 5th, 10th, 15th, 20th, and 25th shapes.
    This means that the 22nd shape is the 2nd shape after the 20th shape, and so is the 2nd shape in the pattern.
    Thus, the 22nd shape is .

    Answer: (A)

  5. The given sum includes 5 terms each equal to (5×5).
    Thus, the given sum is equal to 5×(5×5) which equals 5×25 or 125.

    Answer: (E)

  6. Yihana is walking uphill exactly when the graph is increasing (that is, when the slope of the segment of the graph is positive).
    This is between 0 and 3 minutes and between 8 and 10 minutes, which correspond to lengths of time of 3 minutes and 2 minutes in these two cases, for a total of 5 minutes.

    Answer: (A)

  7. Solution 1

    Since A, B, C, D, E, and F are equally spaced around the circle, moving from one point to the next corresponds to moving 16 of the way around the circle.
    Therefore, moving from A to C corresponds to moving 26 (or 13) of the way around the circle.
    Since moving around the whole circle corresponds to moving through 360°, then moving 13 of the way around the circle corresponds to moving through 13×360°=120°.
    Thus, AOC=120°.

    Solution 2

    We join A, B, C, D, E, and F to O.

    Since A, B, C, D, E, and F are equally spaced around the circle, the angles made at the centre by consecutive points are equal. That is, AOB=BOC=COD=DOE=EOF=FOA Since these 6 angles form a complete circle, the sum of their measures is 360°.
    Therefore, AOB=BOC=COD=DOE=EOF=FOA=16×360°=60° This means that AOC=AOB+BOC=60°+60°=120°.

    Answer: (D)

  8. Since the rectangle has positive integer side lengths and an area of 24, its length and width must be a positive divisor pair of 24.
    Therefore, the length and width must be 24 and 1, or 12 and 2, or 8 and 3, or 6 and 4.
    Since the perimeter of a rectangle equals 2 times the sum of the length and width, the possible perimeters are

    These all appear as choices, which means that the perimeter of the rectangle cannot be 36, which is (E).

    Answer: (E)

  9. Using the definition, 3b=3+b3b.
    Assuming b3, the following equations are equivalent: 3b=43+b3b=43+b=4(3b)3+b=12+4b15=3b and so b=5.

    Answer: (A)

  10. Solution 1

    Since x is 20% of y, then x=20100y=15y.

    Since x is 50% of z, then x=12z.

    Therefore, 15y=12z which gives 25y=z.
    Thus, z=40100y and so z is 40% of y.

    Solution 2 Since x is 20% of y, then x=0.2y.
    Since x is 50% of z, then x=0.5z.
    Therefore, 0.2y=0.5z which gives 0.4y=z.
    Thus, z=0.4y and so z is 40% of y.

    Answer: (D)

  11. The store sells 250 g of jellybeans for $7.50, which is 750 cents.
    Therefore, 1 g of jellybeans costs 750÷250=3 cents.
    This means that $1.80, which is 180 cents, will buy 180÷3=60 g of jellybeans.

    Answer: (C)

  12. Starting with Triangle PQR with side QR vertical, with Q above R, and with the third vertex, P, to the right. and flipping across QR results in Triangle PQR with side QR vertical, with Q above R, and with the third vertex, P, to the left..

    Now flipping across PR results in Triangle PQR with side PQ vertical, with P above Q, and with the third vertex, R, to the right., which is the resulting position that Paola sees.

    Answer: (E)

  13. Since 2×2×2=8, a cube with edge length 2 has volume 8. (We note also that 83=2.)
    Therefore, each of the cubes with volume 8 have a height of 2.
    This means that the larger cube has a height of 2+2=4, which means that its volume is 43=4×4×4=64.

    Answer: (E)

  14. Since 100 000 does not include the block of digits 178, each integer between 10 000 and 100 000 that includes the block of digits 178 has five digits.
    Such an integer can be of the form 178xy or of the form x178y or of the form xy178 for some digits x and y.
    The leading digit of a five-digit integer has 9 possible values (any digit from 1 to 9, inclusive) while a later digit in a five-digit integer has 10 possible values (0 or any digit from 1 to 9, inclusive).
    This means that

    In total, there are thus 100+90+90=280 integers between 10 000 and 100 000 that include the block of digits 178.

    Answer: (A)

  15. Since a+5=b, then a=b5.
    Since a=b5 and c=5+b and b+c=a, then b+(5+b)=b52b+5=b5b=10 (If b=10, then a=b5=15 and c=5+b=5 and b+c=(10)+(5)=(15)=a, as required.)

    Answer: (C)

  16. Extend PQ and TS to meet at point X.

    Rectangle PXTU has point Q on PX, a distance of 10 from P, and point S on XT, a distance of 10 from T. Point R is in the interior of the rectangle with QR perpendicular to PX, RS perpendicular to XT and QR perpendicular to RS. A dashed line from Q to S has length 8.

    Since quadrilateral QRSX has three right angles (at Q, R and S), it must have a fourth right angle at X.
    Thus, QRSX is a rectangle, which means that XS=QR and QX=RS.
    The perimeter of PQRSTU is PQ+QR+RS+ST+TU+UP=PQ+XS+QX+ST+TU+UP=(PQ+QX)+(XS+ST)+TU+UP=PX+XT+TU+UP which is the perimeter of quadrilateral of PXTU.
    But quadrilateral PXTU has four right angles, and so is a rectangle.
    Also, PU=UT, so PXTU is a square, and so the perimeter of PXTU equals 4×PX=4×(PQ+QX)=4×(10+QX)=40+4×QX Finally, QX=PXPQ=PX10=XT10=XTST=XS, which means that QXS is isosceles as well as being right-angled at X.
    By the Pythagorean Theorem, QX2+XS2=QS2 and so 2×QX2=82 or QX2=32.
    Since QX>0, then QX=32=16×2=16×2=42.
    Thus, the perimeter of PQRSTU is 40+4×42=40+16262.6.
    (We could have left this as 40+43262.6.)
    Of the given choices, this is closest to 63.

    Answer: (C)

  17. Zebadiah must remove at least 3 shirts.
    If he removes 3 shirts, he might remove 2 red shirts and 1 blue shirt.
    If he removes 4 shirts, he might remove 2 red shirts and 2 blue shirts.
    Therefore, if he removes fewer than 5 shirts, it is not guaranteed that he removes either 3 of the same colour or 3 of different colours.
    Suppose that he removes 5 shirts.
    If 3 are of the same colour, the requirements are satisfied.
    If no 3 of the 5 shirts are of the same colour, then at most 2 are of each colour. This means that he must remove shirts of 3 colours, since if he only removed shirts of 2 colours, he would remove at most 2+2=4 shirts.
    In other words, if he removes 5 shirts, it is guaranteed that there are either 3 of the same colours or shirts of all 3 colours.
    Thus, the minimum number is 5.

    Answer: (D)

  18. At the beginning of the first day, the box contains 1 black ball and 1 gold ball.
    At the end of the first day, 2 black balls and 1 gold ball are added, so the box contains 3 black balls and 2 gold balls.
    At the end of the second day, 2×2=4 black balls and 2×1=2 gold balls are added, so the box contains 7 black balls and 4 gold balls.
    Continuing in this way, we find the following numbers of balls:

    End of Day # Black Balls Gold Balls
    2 7 4
    3 7+4×2=15 4+4=8
    4 15+8×2=31 8+8=16
    5 31+16×2=63 16+16=32
    6 63+32×2=127 32+32=64
    7 127+64×2=255 64+64=128

    At the end of the 7th day, there are thus 255+128=383 balls in the box.

    Answer: (E)

  19. The line with equation y=2x6 has y-intercept 6.
    Also, the x-intercept of y=2x6 occurs when y=0, which gives 0=2x6 or 2x=6 which gives x=3.
    Therefore, the triangle bounded by the x-axis, the y-axis, and the line with equation y=2x6 has base of length 3 and height of length 6, and so has area 12×3×6=9.
    We want the area of the triangle bounded by the x-axis, the vertical line with equation x=d, and the line with equation y=2x6 to be 4 times this area, or 36.
    This means that x=d is to the right of the point (3,0), because the new area is larger. In other words, d>3.

    Two lines are are drawn on the xy-plane. One line goes up to the right intersecting the y-axis at negative 6 and the x-axis at 3. The other line is vertical with equation x equals d and intersects the x-axis to the right of 3.

    The base of this triangle has length d3, and its height is 2d6, since the height is measured along the vertical line with equation x=d.
    Thus, we want 12(d3)(2d6)=36 or (d3)(d3)=36 which means (d3)2=36.
    Since d3>0, then d3=6 which gives d=9.
    Alternatively, we could note that if similar triangles have areas in the ratio 4:1 then their corresponding lengths are in the ratio 4:1 or 2:1.
    Since the two triangles in question are similar (both are right-angled and they have equal angles at the point (3,0)), the larger triangle has base of length 2×3=6 and so d=3+6=9.

    Answer: (A)

  20. Since 3m3 is a multiple of 3, then 5n5 is a multiple of 3.
    Since 5 is not a multiple of 3 and 3 is a prime number, then n5 is a multiple of 3.
    Since n5 is a multiple of 3 and 3 is a prime number, then n is a multiple of 3, which means that 5n5 includes at least 5 factors of 3.
    Since 5n5 includes at least 5 factors of 3, then 3m3 includes at least 5 factors of 3, which means that m3 is a multiple of 3, which means that m is a multiple of 3.
    Using a similar analysis, both m and n must be multiples of 5.
    Therefore, we can write m=3a5bs for some positive integers a, b and s and we can write n=3c5dt for some positive integers c, d and t, where neither s nor t is a multiple of 3 or 5. (In other words, we have grouped all of the factors of 3 and 5 in each of m and n.)
    From the given equation, 3m3=5n53(3a5bs)3=5(3c5dt)53×33a53bs3=5×35c55dt533a+153bs3=35c55d+1t5 Since s and t are not multiples of 3 or 5, we must have 33a+1=35c and 53b=55d+1 and s3=t5.
    Since s and t are positive and m and n are to be as small as possible, we can set s=t=1, which satisfy s3=t5.
    Since 33a+1=35c and 53b=55d+1, then 3a+1=5c and 3b=5d+1.
    Since m and n are to be as small as possible, we want to find the smallest positive integers a,b,c,d for which 3a+1=5c and 3b=5d+1.
    Neither a=1 nor a=2 gives a value for 3a+1 that is a multiple of 5, but a=3 gives c=2.
    Similarly, b=1 does not give a value of 3b that equals 5d+1 for any positive integer d, but b=2 gives d=1.
    Therefore, the smallest possible values of m and n are m=3352=675 and n=3251=45, which gives m+n=720.
    (We can verify by substitution that m=675 and n=45 satisfy the equation 3m3=5n5.)

    Answer: (C)

  21. Since 20x+11y=881, then 20x=88111y and 11y=88120x.
    Since x is an integer, then 20x is a multiple of 10 and so the units digit of 20x is 0 which means that the units digit of 88120x is 1, and so the units digit of 11y is 1.
    Since the units digit of 11y is 1, then the units digit of y is 1.
    Since 20x is positive, then 11y=88120x is smaller than 881, which means that 11y<881 and so y<8811180.1.
    Thus, the possible values of y are 1, 11, 21, 31, 41, 51, 61, 71.
    We check each of these:

    y 11y=88120x 20x x
    1 11 870 Not an integer
    11 121 760 38
    21 231 650 Not an integer
    31 341 540 27
    41 451 430 Not an integer
    51 561 320 16
    61 671 210 Not an integer
    71 781 100 5

    Therefore, the sum of the smallest and largest of the permissible values of y is 11+71=82.

    Answer: 82

  22. Since the shaded regions are equal, then when the unshaded sector in the small circle is shaded, the area of the now fully shaded sector of the larger circle must be equal to the area of the smaller circle.

    The smaller circle has radius 1 and so has area π×12=π.
    The larger circle has radius 3 and so has area π×32=9π.
    This means that the area of the shaded sector in the larger circle has area π, which means that it must be 19 of the larger circle.
    This means that POQ must be 19 of a complete circle, and so POQ=19×360°=40°.
    Thus, x=40.

    Answer: 40

  23. Suppose that Andreas, Boyu, Callista, and Diane choose the numbers a, b, c, and d, respectively.
    There are 9 choices for each of a, b, c, and d, so the total number of quadruples (a,b,c,d) of choices is 94=6561.
    Among the 9 choices, 5 are odd (1,3,5,7,9) and 4 are even (2,4,6,8).
    If there are N quadruples (a,b,c,d) with a+b+c+d even (that is, with the sum of their choices even), then the probability that the sum of their four integers is even is N6561, which is in the desired form.
    Therefore, we count the number of quadruples (a,b,c,d) with a+b+c+d even.
    Among the four integer a, b, c, d, either 0, 1, 2, 3, or 4 of these integers are even, with the remaining integers odd.
    If 0 of a, b, c, d are even and 4 are odd, their sum is even.
    If 1 of a, b, c, d is even and 3 are odd, their sum is odd.
    If 2 of a, b, c, d are even and 2 are odd, their sum is even.
    If 3 of a, b, c, d are even and 1 is odd, their sum is odd.
    If 4 of a, b, c, d are even and 0 are odd, their sum is even.
    Therefore, we need to count the number of quadruples (a,b,c,d) with 0, 2 or 4 even parts.

    If 0 are even and 4 are odd, there are 5 choices for each of the parts, and so there are 54=625 such quadruples.
    If 4 are even and 0 are odd, there are 4 choices for each of the parts, and so there are 44=256 such quadruples.
    If 2 are even and 2 are odd, there are 4 choices for each of the even parts and 5 choices for each of the odd parts, and 6 pairs of locations for the even integers (ab,ac,ad,bc,bd,cd) with the odd integers put in the remaining two locations after the locations of the even integers are chosen. Thus, there are 42526=2400 such quadruples.
    In total, this means that there are 625+256+2400=3281 quadruples and so N=3281.
    The sum of the squares of the digits of N is equal to 32+22+82+12=9+4+64+1=78.

    Answer: 78

  24. Let O be the vertex of the cube farthest away from the table.
    Let A, B and C be the vertices of the cube connected to O with edges.
    Since the cube has edge length 8, then OA=OB=OC=8.
    Note that AOB=AOC=BOC=90°, which means that each of AOB, AOC and BOC is a right-angled isosceles triangle, which means that AB=2AO=82, and so AB=AC=BC=82.
    Let P, Q and R be the vertices that complete the square faces PAOB, QAOC and RBOC.
    From directly above, the cube looks like this:

    When the sun is directly overhead, the shadow of the cube will look exactly like the area of the “flat” hexagon APBRCQ. In mathematical terms, we are determining the area of what is called a projection.
    To find the area of this figure, we need to know some lengths, but we have to be careful because not all of the edges in the edges in the diagram above are “flat”.
    We do know that points A, B and C are in the same horizontal plane, and we also know that AB=AC=BC=82. This means that these are true lengths in that the points A, B and C are this distance apart.
    Note that flat quadrilateral PAOB is divided into two regions of equal area by AB. Similarly, QAOC is divided into two regions of equal area by AC, and RBOC is divided into two regions of equal area by BC.
    In other words, the area of ABC is half of the area of hexagon APBRCQ, so to find the area of APBRCQ, we double the area of ABC.
    So we need to calculate the area of equilateral ABC which has side length 82.
    Let M be the midpoint of BC. Thus, BM=CM=42.

    Since ABC is equilateral, then AM is perpendicular to BC.
    By the Pythagorean Theorem in AMC, AM=AC2MC2=(82)2(42)2=12832=96 Therefore, the area of ABC is 12CBAM=128296=4192=4643=483=323 This means that the area of the hexagonal shadow is 643.
    Since this is in the form ab where a and b are positive integers and b is not divisible by the square of an integer larger than 1, then a=64 and b=3 and so a+b=67.

    Answer: 67

  25. We begin by tracing what happens when T=337.
    We start with tokens labelled 1,2,3,,335,336,337 arranged around a circle.
    We remove the first token (1), move 2 tokens along and remove that token (3), move 2 tokens along and remove that token (5), and continue around the circle until we remove tokens 335 and 337.
    This leaves tokens 2,4,6,,332,334,336. These tokens differ by 2.

    On the second pass, we start with tokens labelled 2,4,6,,332,334,336, which differ by 2.
    Because the last token was removed on the first pass (337), the first token is not removed on the second pass, which means that we remove every other token starting with 4.
    This means that the remaining tokens differ by 4, and are 2,6,10,,326,330,334.

    On the third pass, we start with 2,6,10,,326,330,334, which differ by 4.
    Because the last token (336) was removed on the previous pass, we remove every other token starting with 6.
    The remaining tokens differ by 8 and are 2,10,18,,314,322,330.

    On the fourth pass, we start with 2,10,18,,314,322,330, which differ by 8.
    Because the last token (334) was removed on the previous pass, we remove every other token starting with 10.
    The remaining tokens differ by 16 and are 2,18,34,,290,306,322.

    On the fifth pass, we start with 2,18,34,,290,306,322, which differ by 16.
    Because the last token (330) was removed on the previous pass, we remove every other token starting with 18.
    The remaining tokens differ by 32 and are 2,34,66,98,130,162,194,226,258,290,322.

    On the sixth pass, we start with 2,34,66,98,130,162,194,226,258,290,322, which differ by 32.
    Because the second last token (306) was removed on the previous pass, we remove ever other token starting with 2.
    This leaves 34,98,162,226,290, which differ by 64.

    On the seventh pass, we remove starting with the second token, which leaves 34,162,290, which differ by 128.

    On the eighth pass, we remove starting with the first token, which leaves 162.

    This tells us that the smallest possible value of T is at least 162 and at most 337.

    Next, we will show that T=209 also gives a final token of 162 by working through the various passes. (This is a place where how we discover the answer is more than likely different than how we justify the answer.)
    Before first pass: 1,2,3,,207,208,209 After first pass (removing every other token starting with 1): 2,4,6,,204,206,208 After second pass (removing every other token starting with 4): 2,6,10,,198,202,206 After third pass (removing every other token starting with 6): 2,10,18,,186,194,202 After fourth pass (removing every other token starting with 10): 2,18,34,,162,178,194 After fifth pass (removing every other token starting with 18): 2,34,66,98,130,162,194 After sixth pass (removing every other token starting with 2): 34,98,162 After seventh pass (removing 98): 34,162 This leaves 162 after the eighth pass, since 34 will be removed.
    Therefore, when T=209 and when T=337, the final token is 162.

    Finally, we show that if the final token starting with T tokens is 162, then T209, which will tell us that the smallest value of T is 209.
    Suppose that T209 and that the token remaining after the final pass is 162.
    Before each pass, the remaining tokens differ by a power of 2, since we start by removing every other token from a list that differs by 1, then every other token from a list that differs by 2, and so on.
    The smallest powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256.

    Since 162 is left after the last pass (this will turn out to be the eighth pass), the remaining tokens must have differed by 128 before the eighth pass, and thus were 34, 162. (Since T209, then there could not be a token numbered 162+128=290.) Also, if the tokens differed by 64 before the eighth pass, there would have been tokens labelled 34 and 98 that were both removed.
    Thus, before the eighth pass, the tokens were 34, 162 and 34 was removed.

    Before the seventh pass, the tokens differed by 64.
    Thus, these were 34, 98, 162. We note that the last token was not removed on this pass, and so the first token is removed on the eighth pass, as expected.
    Also, there cannot be a token numbered 162+64=226, since T209.

    Before the sixth pass, the tokens differed by 32.
    Thus, these were 2, 34, 66, 98, 130, 162, 194. The last token cannot have been 162 since the last token must be removed on this pass so that the second token (98) is removed on the seventh pass. Thus, 162+32=194 must be the last token here. (Note that 226 was already rejected earlier.)

    Before the fifth pass, the tokens differed by 16.
    Since the first token (2) is removed on the sixth pass, the last token is not removed on the fifth pass.
    This means that the tokens before this pass were 2,18,34,50,66,82,98,114,130,146,162,178,194 On this pass, 178 is the last token removed. (Note that 194+16=210 is too large.)

    Before the fourth pass, the tokens differed by 8.
    Since the second token (18) is removed on the fifth pass, the last token is removed on the fourth pass.
    This means that the tokens before this pass were 2,10,18,,162,170,178,186,194,202 The token 202 must be included since 194 remains for the fifth pass. (Note that 202+8=210 is too large.)

    Before the third pass, the tokens differed by 4.
    Since the second token (10) is removed on the fourth pass, the last token is removed on the third pass.
    This means that the tokens before this pass were 2,6,10,14,18,,186,190,194,198,202,206 The token 206 must be included since 202 remains for the fourth pass. (Note that 206+4=210 is too large.)

    Before the second pass, the tokens differed by 2.
    Since the second token (6) is removed on the third pass, the last token is removed on the second pass.
    This means that the tokens before this pass were 2,4,6,8,,198,200,202,204,206,208 The token 208 must be included since 206 remains for the third pass. (Note that 208+2=210 is too large.)

    Before the first pass, the tokens differed by 1.
    Since the second token (4) is removed on the second pass, the last token is removed on the first pass.
    This means that the tokens before this pass were 1,2,3,4,,204,205,206,207,208,209 The token 209 must be included since 208 remains for the second pass. (Note that 209+1=210 is too large.)

    Therefore, we must have at least 209 tokens for the final token to be 162, and so the smallest possible value of T is 209, whose rightmost two digits are 09.

    Answer: 09


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