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2023 Hypatia Contest
Solutions
(Grade 11)

Wednesday, April 5, 2023
(in North America and South America)

Thursday, April 6, 2023
(outside of North American and South America)

©2023 University of Waterloo


    1. Jasmin’s total score was 32+45=6+20 or 26 points.

    2. Suppose that Sam had n throws that each scored 5 points; then Sam had 2n throws that each scored 2 points.
      Since Sam’s total score was 36 points, then 5n+22n=36 or 9n=36, and so n=4.
      In total, Sam took n+2n=3n throws, which is 34=12 throws.

    3. Since Tia’s total score was 37 points, then 2t+5f=37.
      Since 2t is even for all integer values of t, then 5f must be odd since their sum is 37 (which is odd).
      The value of 5f is odd exactly when f is odd.
      When f=1, we get 2t+5=37 or 2t=32, and so t=16.
      When f=3, we get 2t+15=37 or 2t=22, and so t=11.
      When f=5, we get 2t+25=37 or 2t=12, and so t=6.
      When f=7, we get 2t+35=37 or 2t=2, and so t=1.
      When f9, 5f45 and so 2t+5f>37.
      The possible ordered pairs (t,f) are (16,1), (11,3), (6,5), and (1,7).

    4. If a throws each score 6 points and b throws each score 21 points, then 6a+21b or 3(2a+7b) points are scored.
      Since a and b are non-negative integers, then 2a+7b is a non-negative integer, and so the total number of points scored, 3(2a+7b), is a multiple of 3.
      Since 182 is not a multiple of 3, then it is not possible to have a total score of 182 points.

    1. The total area of the shaded regions can be determined by subtracting the area of AED from the area of rectangle ABCD.
      The area of rectangle ABCD is 215=30.
      The base length of AED is AD=15, its height is equal to AB=2, and so its area is 12152=15.
      Thus, the total area of the shaded regions is 3015=15.

    2. Solution 1

      Since AFD has base length AD=15 and height equal to AB=2, then its area is 12152=15.

      The area of AFD is equal to the sum of the areas of AGD and FGD.

      Since the area of AFD is 15, and the area of FGD is 5, then the area of AGD is 155=10.
      The area of ABD is 12152=15.
      The area of ABD is equal to the sum of the areas of AGD and ABG.
      Since the area of ABD is 15, and the area of AGD is 10, then the area of ABG (the shaded area) is 1510=5.

      Solution 2

      Consider AFD and ABD. Each has base AD and height AB and thus the two triangles have equal areas.
      The area of AFD is equal to the sum of the areas of AGD and FGD.
      The area of ABD is equal to the sum of the areas of AGD and ABG.
      Thus, the area of ABG (the shaded area) is equal to the area of FGD, which is 5.

    3. The total area of the bottom two shaded regions (ASR and RQD) can be determined by subtracting the area of PQRS from the area of APD.

      The base length of APD is AD=15, its height is equal to AB=2, and so its area is 12152=15.
      Since the area of PQRS is 6, the total area of the two bottom shaded regions is 156=9.

      We can similarly show that the total area of the two top shaded regions (BSP and PQC) can be determined by subtracting the area of PQRS from the area of BRC.
      Since BRC also has area 15, then the total area of the two top shaded regions is also 156=9.
      Thus, the total area of the shaded regions is 9+9=18.

    1. Switching the second and fourth digits of 6238 gives the cousin 6832.
      Since 6832 is not in the list, then it is the missing cousin.

    2. Of the 16 six-digit integers in the given list, 15 are cousins of the original integer.
      If the original six-digit integer is abcdef, then there are exactly 5 cousins for which a is not the 1st digit.
      These 5 cousins are a result of switching the 1st digit, a, with each of the other 5 digits, b,c,d,e,f, and thus are bacdef, cbadef, dbcaef, ebcdaf, and fbcdea.
      Each of the remaining 155=10 cousins of abcdef have first digit a.
      Since the digits are distinct, then the 1st digit of the original integer is the integer that occurs most frequently as the 1st digit in the given list, which is 7.
      A similar argument can be made for each of the other digits.
      That is, the nth digit of the original integer is the integer that occurs most frequently as the nth digit in the given list.
      Thus, the original integer has 2nd digit 2, 3rd digit 6, 4th digit 4, 5th digit 9, 6th digit 1, and so the original integer is 726 491.
      We may check that if the original integer is 726 491, then the cousins are indeed the remaining 15 integers in the given list.

    3. The cousins of the three-digit integer cd3 are dc3, 3dc and c3d.
      The difference between the two integers cd3 and dc3 could be negative, in which case it’s not the right difference. If it is positive, then cd3 minus dc3 has ones digit 0, and thus cannot equal d95.
      Similarly, the difference between the two integers cd3 and c3d could be negative, in which case it’s not the right difference. Since cd3 and c3d each have hundreds digit c, then cd3 minus c3d has hundreds digit 0, and thus cannot equal d95 (since d is a non-zero digit).
      Therefore, cd3 minus 3dc is equal to d95.
      Since the difference, d95, has ones digit 5, then c=8. (You should confirm for yourself that this is the only possible value for c before moving on.)
      With c=8, we get 8d3 minus 3d8 is equal to d95.
      The three-digit integer 8d3 is equal to 800+10d+3.
      The three-digit integer 3d8 is equal to 300+10d+8.
      Thus, 8d3 minus 3d8 is equal to (800+10d+3)(300+10d+8)=5005=495, which is equal to d95, and so d=4.
      (We may confirm that 843348=495.)
      The value of c is 8 and the value of d is 4, and no other values are possible.

    4. Solution 1

      The six cousins of the four-digit integer mn97 are nm97, 9nm7, 7n9m, m9n7, m79n, and mn79.
      The cousin nm97 is equal to 1000n+100m+90+7.
      The cousin 9nm7 is equal to 9000+100n+10m+7.
      The cousin 7n9m is equal to 7000+100n+90+m.
      The cousin m9n7 is equal to 1000m+900+10n+7.
      The cousin m79n is equal to 1000m+700+90+n.
      The cousin mn79 is equal to 1000m+100n+70+9.
      The sum, S, of the six cousins is thus S=1000(n+9+7+m+m+m)+100(m+n+n+9+7+n)+10(9+m+9+n+9+7)+(7+7+m+7+n+9)=1000(3m+n+16)+100(m+3n+16)+10(m+n+34)+(m+n+30) The sum, S, must be equal to the five-digit integer nmnm7, which has ones digit 7.
      The ones digit of S=1000(3m+n+16)+100(m+3n+16)+10(m+n+34)+(m+n+30) is equal to the ones digit of m+n+30, which is equal to the ones digit of m+n (since the ones digit of 30 is 0).
      If the ones digit of m+n is 7, then m+n=7 or m+n=17.
      Since m and n are distinct non-zero digits, then the possible pairs (m,n) are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (8,9), and (9,8).
      If (m,n)=(1,6), then the five-digit integer nmnm7 is 61 617, and S=1000(3m+n+16)+100(m+3n+16)+10(m+n+34)+(m+n+30)=1000(3(1)+6+16)+100(1+3(6)+16)+10(1+6+34)+(1+6+30)=1000(25)+100(35)+10(41)+37=28947 and so (m,n)=(1,6) is not a possible pair.
      We continue to check the remaining possible pairs (m,n) in the table below.

      (m,n) nmnm7 S
      (2,5) 52 527 30 747
      (3,4) 43 437 32 547
      (4,3) 34 347 34 347
      (5,2) 25 257 36 147
      (6,1) 16 167 37 947
      (8,9) 98 987 54 657
      (9,8) 89 897 56 457

      The only pair of distinct, non-zero digits (m,n) is (4,3).

      Solution 2

      The six cousins of the four-digit integer mn97 are nm97, 9nm7, 7n9m, m9n7, m79n, and mn79. As in Solution 1, the sum, S, of the six cousins is S=1000(n+9+7+m+m+m)+100(m+n+n+9+7+n)+10(9+m+9+n+9+7)+(7+7+m+7+n+9)=1000(3m+n+16)+100(m+3n+16)+10(m+n+34)+(m+n+30)=3000m+1000n+16000+100m+300n+1600+10m+10n+340+m+n+30=3111m+1311n+17970 The sum, S, must be equal to the five-digit integer nmnm7, which is equal to 10000n+1000m+100n+10m+7 or 10100n+1010m+7.
      Setting these two expressions equal to one another and simplifying, we get 3111m+1311n+17970=10100n+1010m+78789n2101m=17963799n191m=1633(after division by 11) Since 799n=1633+191m and m1, then 799n1633+191(1) or 799n1824, and so n18247992.3.
      Since m9, then 799n1633+191(9) or 799n3352, and so n33527994.2.
      Since 2.3n4.2 and n is an integer, then the possible values of n are 3 and 4.
      Substituting n=4, we get 799(4)=1633+191m or 191m=1563, and since 1563191 is not an integer, then n4.
      Substituting n=3, we get 799(3)=1633+191m or 191m=764, and so m=4.
      When m=4 and n=3, the five-digit integer nmnm7 is 34 347 and S=3111(4)+1311(3)+17970 or S=34347, and so the only pair of distinct, non-zero digits (m,n) is (4,3).

    1. Since there are 9 ways to generate each of the three integers, there are a total of 999=729 ways to generate all three integers.
      Next, we consider the different cases for which the product of the three integers is a prime number.
      The integers from 1 to 9 include the composite numbers 4, 6, 8, and 9.
      If any one of the three integers generated is composite, then the product of the three integers is composite.
      Thus, the integers must be chosen from 1, 2, 3, 5, and 7.
      Of these five integers, 2, 3, 5, and 7 are prime numbers.
      If two or more of the three integers generated are prime numbers, then the product of the three integers is composite.
      Thus, at most one integer is a prime number.
      If no integer generated is a prime number (all three are equal to 1), then the product is 1, which is not prime.
      Thus, Amarpreet’s product is a prime number exactly when one of the integers is 2, 3, 5, or 7, and each of the other two integers is equal to 1.
      There are 4 ways to choose one of the prime numbers, p, and 3 ways to arrange the integers 1, 1, p, and so there are 43=12 ways to generate three integers whose product is a prime number.
      Therefore, the probability that the product is a prime number is 12729=4243.

    2. Solution 1

      We begin by considering the different cases for which the product of the four integers is divisible by 5, but not divisible by 7.
      The only integer from 1 to 9 that is divisible by 5 is 5, and so the product of the four integers is divisible by 5 exactly when at least one of the four integers is a 5.
      Similarly, the only integer from 1 to 9 that is divisible by 7 is 7, and so the product of the four integers is not divisible by 7 exactly when a 7 is not among the four integers generated.
      We proceed to count the number of ways to generate such arrangements of four integers by considering the number of times that 5 appears in the arrangement.

      Case 1: there are four 5s

      Since each of the four integers must be 5, there is 1 such possibility in this case.

      Case 2: there are exactly three 5s

      The three 5s can be arranged in 4 different ways: 555_, 55_5, 5_55, _555.
      There are 7 choices for the integer that is not a 5 since it can be any of the nine integers, except 5 and 7.
      Thus, there are 47=28 possibilities in this case.

      Case 3: there are exactly two 5s

      The two 5s can be arranged in 6 different ways: 55__, 5_5_, 5__5, _55_, _5_5, __55.
      Once the two 5s have been placed, there are 7 choices for each of the remaining two integers (since it is possible that these integers are equal to one another).
      Thus, there are 677=294 possibilities in this case.

      Case 4: there is exactly one 5

      The 5 can be placed in 4 different ways.
      Once the 5 has been placed, there are 7 choices for each of the remaining three integers (since it is possible that these integers are equal to one another).
      Thus, there are 4777=1372 possibilities in this case.

      In total, there are 1+28+294+1372=1695 ways to select the four integers.
      Since there are 9 ways to generate each of the four integers, there are a total of 94=6561 ways to generate all four integers.
      Therefore, the probability that Braxton’s product is divisible by 5, but not divisible by 7 is 16956561=5652187.

      Solution 2

      Let A be the event that the product generated by Braxton is divisible by 5, and A be the event that the product is not divisible by 5.
      Let B be the event that the product generated by Braxton is divisible by 7, and B be the event that the product is not divisible by 7.
      We are asked to find the probability of A and B, which we write as P(A and B).
      Consider the Venn diagram shown in Figure 1.
      The shaded region of this diagram represents P(A and B).

      Figure 1 is a Venn diagram containing two
overlapping circles enclosed in a rectangle labelled P(A and B
overline). The left circle is labelled P(A) and is shaded except for the
overlapping region with the circle on the right. The right circle is
labelled P(B) and the complete circle is unshaded. The region outside of
the circles and inside the rectangle is unshaded.

      We wish to determine a more efficient method for determining P(A and B) than that which was shown in Solution 1.
      To do so, we proceed to use Venn diagrams to help express P(A and B) in an equivalent form.

      Consider the next two Venn diagrams shown below.
      In Figure 2, the shaded region represents P(B).
      In Figure 3, the shaded region represents P(A and B).

      Figure 2 is a Venn diagram containing two
overlapping circles enclosed in a rectangle labelled P(B overline). The
left circle is labelled P(A) and is shaded except for the overlapping
region with the circle on the right. The right circle is labelled P(B)
and the complete circle is unshaded. The region outside the circles and
inside the rectangle is shaded. Figure 3 is a Venn
diagram containing two overlapping circles enclosed in a rectangle
labelled P(A overline and B overline). The left circle is labelled P(A)
and is completely unshaded. The right circle is labelled P(B) and is
completely unshaded including region of overlap. The region outside the
circles and inside the rectangle is shaded.

      Notice that if the region that is shaded in Figure 3 is removed (unshaded) in Figure 2, then the resulting Venn diagram is equivalent to that in Figure 1.
      That is, mathematically, P(A and B)=P(B)P(A and B).
      (Without the use of the Venn diagrams, we may have similarly noticed that since exactly one of A and A occurs, then P(B)=P(A and B)+P(A and B), and so P(B)P(A and B)=P(A and B)+P(A and B)P(A and B)=P(A and B). )

      Thus, the probability that the product is divisible by 5, but not divisible by 7, is equal to the probability that the product is not divisible by 7, minus the probability that the product is both not divisible by 5 and not divisible by 7.
      The only integer from 1 to 9 that is divisible by 7 is 7, and so the product of the four integers is not divisible by 7 exactly when a 7 is not among the four integers generated.
      The probability that the integer generated is not 7 is 89, and so the probability that the product of the four integers is not divisible by 7 is P(B)=(89)4.
      The only integer from 1 to 9 that is divisible by 5 is 5, and so the product of the four integers is not divisible by 5 exactly when a 5 is not among the four integers generated.
      The probability that the integer generated is not divisible by both 5 and 7 is thus 79, and so the probability that the product of the four integers is not divisible 5 and not divisible by 7 is P(A and B)=(79)4.
      Therefore, the probability that the product is divisible by 5, but not divisible by 7, is (89)4(79)4=847494=409624016561=16956561=5652187.

      Solution 3

      Suppose that the four numbers generated are a, b, c, d.
      When there are no restrictions, there are 9 choices for each of a, b, c, d, and so a total of 94 possible combinations of 4 digits that can be generated.
      If abcd is not divisible by 7, then none of a, b, c, d can be 7; thus, there are 8 possibilities for each of a, b, c, d, and so 84 of these 94 combinations give products that are not divisible by 7.
      If abcd is not divisible by 7 and not divisible by 5, then none of a, b, c, d can be 5; thus, there are 7 possibilities for each of a, b, c, d, and so 74 of these 84 combinations whose products are not divisible by 7 give products that are also not divisible by 5.
      This means that there are 8474 combinations whose product is not divisible by 7 but whose product is divisible by 5.
      Therefore, the probability that a random combination gives a product that is not divisible by 7 but is divisible by 5 is 847494=409624016561=16956561=5652187.

    3. If any one of the 2023 integers generated is a 6, then the product is divisible by 6.
      Thus, we exclude 6 from the choice of possible integers.
      Further, an integer is divisible by 6 exactly when it is divisible by both 2 and 3.
      With 6 excluded, the remaining integers that are divisible by 2 are 2, 4 and 8, and the remaining integers that are divisible by 3 are 3 and 9.
      Therefore, the product is also divisible by 6 when at least one of the integers generated is 2 or 4 or 8, and at least one of the integers generated is 3 or 9.
      To summarize, the product is not divisible by 6 exactly when

      1. 6 is not among the integers generated, and

      2. there are integers from at most one of the two lists 2, 4, 8, and 3, 9.

      Thus, there are exactly 3 cases which produce a product that is not divisible by 6, as follows.

      Case Is there a 6? Is there a 2, 4 or 8? Is there a 3 or 9?
      1 no no no
      2 no no yes
      3 no yes no

      For each of these 3 cases, we count the number of ways that the product generated is not divisible by 6.

      Case 1: there is no 6, 2, 4, 8, 3, or 9

      In this case, there are 3 integers that may be chosen (1, 5 and 7), and so there are 32023 ways to generate this product.

      Case 2: there is no 6, 2, 4, or 8

      In this case, there are 5 integers that may be chosen (1, 3, 5, 7, and 9), and so there are 52023 ways to generate this product.
      However, this count includes the cases in which only 1, 5 and 7 are chosen (3 and 9 are not), which were counted in Case 1.
      Thus, the number of ways to generate the products in Case 2, different from those in Case 1, is 5202332023.

      Case 3: there is no 6, 3, or 9

      In this case, there are 6 integers that may be chosen (1, 2, 4, 5, 7, and 8), and there are 62023 ways to generate this product.
      However, this count includes the cases in which only 1, 5 and 7 are chosen (2, 4 and 8 are not), which were counted in Case 1.
      Thus, the number of ways to generate the products in Case 3, different from those in Case 1, is 6202332023.

      Thus, the total number of ways to generate all products that are not divisible by 6 is 32023+5202332023+6202332023=52023+6202332023 When there are no restrictions, there are 9 choices for each integer generated, and so there is a total of 92023 possible ways to generate the product.
      Thus, the probability, p, that the product is not divisible by 6 is p=52023+620233202392023 and so p92023=52023+6202332023, which is equal to the sum and difference of integers and thus an integer.
      To determine the ones digit of the integer equal to p92023, we determine the ones digit of the integer equal to 52023+6202332023.
      The ones digit of 5a is equal to 5 for all positive integers a, and thus the ones digit of 52023 is 5.
      The ones digit of 6b is equal to 6 for all positive integers b, and thus the ones digit of 62023 is 6.
      Consider the ones digit of successive integer powers of 3 beginning at 31:

      • 31 has ones digit 3

      • 32 has ones digit 9

      • 33 has ones digit 7

      • 34 has ones digit 1

      • 35 has ones digit 3

    Since 31 and 35 have the same ones digit and we are performing the same action to get from one step to the next, the results begin to repeat. Thus the block of 4 ones digits (3, 9, 7, 1) must form a cycle.
    Since 2023=4505+3, then the ones digit of 32023 is the third number in the repeating block, which is 7.
    Finally, the ones digit of the integer equal to p92023 is the ones digit of 5+67, which is equal to 4.