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2018 Galois Contest
Solutions
(Grade 10)

Thursday, April 12, 2018
(in North America and South America)

Friday, April 13, 2018
(outside of North American and South America)

©2018 University of Waterloo


    1. Simplifying, we get 12x23x=4x for x0.

    2. Since 12x23x=4x, the value of the expression 12x23x is equal to the value of the simplified expression 4x for all values of x0.

      So when x=5, the value of the expression 12x23x is equal to 4(5)=20.

    3. Simplifying, we get 8mn3m2=8n3m for m0.

      The value of the expression 8mn3m2 is equal to the value of the simplified expression 8n3m for all values of m0.

      Substituting n=2m into 8n3m and simplifying, we get 8(2m)3m=16m3m=163.

      When n=2m and m0, the value of the expression 8mn3m2 is 163.

    4. Simplifying, we get 8p2q5pq2=8p5q for p0, q0.

      When q=6, we get 8p5q=8p5(6)=8p30=4p15.

      That is, when q=6 (and p0) the expression 8p2q5pq2 is equal to 4p15, and so the solution to 38p2q5pq24 is equivalent to the solution to 34p154.

      Solving 34p154, we get 454p60 or 454p604, and so 11.25p15.

      Since p is a positive integer, then p=12,13,14,15.

      Note: In each of the solutions to (b), (c) and (d), we chose to simplify the expression before substituting. Changing the order to substitution followed by simplification would also allow us to solve these problems.

    1. In ABC, ABC=90.

      Using the Pythagorean Theorem, we get AC2=AB2+BC2 or AC2=82+152, and so AC=64+225=289=17 (since AC>0).

    2. In Figure 2, EF is a diameter and so its length is twice the radius or 26.

      From the second fact, we know that EDF=90.

      Using the Pythagorean Theorem, we get DF2=EF2DE2 or DF2=262242, and so DF=676576=100=10 (since DF>0).

    3. Since SQ is a diameter, then SPQ=SRQ=90.

      In SPQ, SP=PQ which means that SPQ is isosceles and so

      PQS=PSQ=180902=45.

      Since RQP=80, then RQO=RQPPQS=8045=35.

      In ROQ, OR=OQ (both are radii) and so QRO=RQO=35 and

      ROQ=1802×35=110.

      In SRQ, we get RSQ=180SRQRQS=1809035=55.

    1. A cylinder having radius r and height h has volume πr2h.

      Cylinder A has radius 12 and height 25, and so its volume is π(12)2(25)=3600π.

      Before Cylinder B is lowered into Cylinder A, the height of water in Cylinder A is 19, and so initially the volume of water in Cylinder A is π(12)2(19)=2736π.

      The height of Cylinder B, 30, is greater than the height of Cylinder A, and so it is not possible for water to pour out of Cylinder A and into Cylinder B.

      When Cylinder B is lowered to the bottom of Cylinder A, the portion of Cylinder B lying inside Cylinder A has radius 9 and height 25 (the height of Cylinder A).

      Thus, the volume that Cylinder B occupies within Cylinder A is π(9)2(25)=2025π.

      Since water cannot pour into Cylinder B, the space available for water within Cylinder A (and outside Cylinder B) is the difference between the volume of Cylinder A and the volume of Cylinder B lying inside Cylinder A, or 3600π2025π=1575π.

      The volume of water in Cylinder A was initially 2736π and once Cylinder B is lowered to the bottom of Cylinder A, the space available for water in Cylinder A becomes 1575π.

      Therefore, the volume of water that spills out of Cylinder A and onto the ground is 2736π1575π=1161π.

    2. As Cylinder B is lowered into Cylinder A, water spills out of Cylinder A and onto the ground when:

      1. the volume of water in Cylinder A exceeds the volume inside Cylinder A and outside Cylinder B, and

      2. the top of Cylinder B lies above the top of Cylinder A.

        (See Figure 1 given in the question.)

      As Cylinder B is lowered into Cylinder A, water spills out of Cylinder A and into Cylinder B when:

      1. the top of Cylinder B lies below the top of Cylinder A, and

      2. the volume of water in Cylinder A (and outside Cylinder B) exceeds the volume inside Cylinder A that lies below the top of Cylinder B and outside Cylinder B, and

      3. Cylinder B is not full of water.

        (See Figure 2 given in the question.)

      In Figure 3 shown, the top of Cylinder B has been lowered to the same level as the top of Cylinder A.

      At this point, the volume of space inside Cylinder A and outside Cylinder B is π(12)2(25)π(9)2(20)=3600π1620π=1980π.

      The initial volume of water in Cylinder A was 2736π, and so at this point the volume of water that has spilled out of Cylinder A and onto the ground is 2736π1980π=756π.

      (Since the top of Cylinder B is not below the top of Cylinder A, no water has spilled out of Cylinder A and into Cylinder B at this point.)

      As Cylinder B is lowered below this level, water will spill out of Cylinder A and into Cylinder B. How much water will spill into Cylinder B?

      Figure 4. A cross sectional view of a smaller cylinder placed inside a larger cylinder, with the tops of the two cylinders at the same level. In the larger cylinder, the space not taken up by the smaller cylinder is completely filled with water. The smaller cylinder is empty. The height of the larger cylinder is 25. The height of the smaller cylinder is 20. The water in the larger cylinder that is directly below the base of the smaller cylinder is labelled U. The height of this water is 5.

      In Figure 4, the volume of water labelled U (lying directly underneath Cylinder B) will be displaced by Cylinder B when it is lowered to the bottom of Cylinder A.

      This volume of water will spill into Cylinder B (since the top of Cylinder B will be below the top of Cylinder A).

      The shape of the water labelled U is cylindrical, has radius equal to that of Cylinder B, 9, and has height 2520=5.

      So the volume of the water labelled U is π(9)2(5)=405π.

      Figure 5. The base of the smaller cylinder lies on the base of the larger cylinder. The top of the smaller cylinder lies below the top of the larger cylinder. Dashed lines extend the vertical walls of the smaller cylinder to meet the top of the larger cylinder. The region in the larger cylinder that lies above the top of the smaller cylinder and outside the (extended) vertical walls smaller cylinder is labelled S. The height of this region is 5.

      In addition, the water labelled S in Figure 5 will also spill into Cylinder B when it is lowered to the bottom of Cylinder A.

      The shape of the water labelled S is a cylindrical ring, inside Cylinder A and outside Cylinder B, having height 2520=5, and so has volume π(12)2(5)π(9)2(5)=315π.

      The volume of water that spills from Cylinder A into Cylinder B is 405π+315π=720π.

      The depth, d, of water in Cylinder B when it is on the bottom of Cylinder A is given by π(9)2(d)=720π and so d=720π81π=809.

      Figure 6. The tops of the two cylinders are at the same level. Dashed lines extend the vertical walls of the smaller cylinder to meet the bottom of the larger cylinder. A dashed line extends the base of the smaller cylinder to meet the vertical walls of the larger cylinder. The region in the larger cylinder that lies below the (extended) base of the smaller cylinder and outside the (extended) vertical walls of the smaller cylinder is labelled S.

      Note: We could have determined the volume of water that spills into Cylinder B by noticing that the volume labelled S (in Figure 5), is equal to the volume of water surrounding the water labelled U (see Figure 6).

      Since both volumes have height 5, their combined volume is equal to that of a cylinder with radius 12 and height 5, and so V=π(12)2(5)=720π, as we previously determined.

    3. Solution 1

      We begin by finding the range of values of h for which some water will spill out of Cylinder A when Cylinder B is lowered to the bottom of Cylinder A.

      Consider lowering Cylinder B into Cylinder A until the water level reaches the top of Cylinder A, as shown in Figure 7 (we know this is possible for some values of h since it occured in part (a)).

      Let y be the distance between the bottoms of the two cylinders, and so the distance between the top of Cylinder A and the bottom of Cylinder B is 25y.

      Figure 7

      The volume of water, Vw, is equal to the volume of Cylinder A that lies below the bottom of Cylinder B, or π(12)2(y), added to the volume inside Cylinder A and outside Cylinder B between the top of Cylinder A and the bottom of Cylinder B, orπ(12)2(25y)π(9)2(25y)=π(12292)(25y).

      That is, Vw=π(12)2(y)+π(12292)(25y)=144πy+63π(25y)=81πy+1575π.

      From part (a), the initial volume of water is 2736π, and so we get81πy+1575π=2736π or 81πy=1161π, and so y=433.

      So if h>25y=25433=323, then water will spill out of Cylinder A onto the ground.

      What if h323?

      When h323, Cylinder B may be lowered so that its top is level with the top of Cylinder A without any water spilling out of Cylinder A onto the ground.

      In this case when h323, then y25323=433, and so y>h.

      That is, when Cylinder B is lowered so that its top is level with the top of Cylinder A, the volume of water that lies directly below Cylinder B is greater that the volume of Cylinder B and so Cylinder B will be completely full of water when it is lowered to the bottom of Cylinder A.

      In this question, we require that Cylinder B not be full and so h>323 and water will spill out of Cylinder A onto the ground before the top of Cylinder B is level with the top of Cylinder A.

      Next, we will further restrict the range of values of h so that when Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full.

      Consider lowering Cylinder B to the point where the tops of the two cylinders are level with one another (so then h25).

      Some water has spilled out of Cylinder A.

      When Cylinder B is lowered beyond this point (so then we require h<25), water will spill from Cylinder A into Cylinder B (and not onto the ground).

      Figure 8

      From the solution in part (b), recall that when Cylinder B is lowered to the bottom of Cylinder A, the volume of water that will spill from Cylinder A into Cylinder B is equal to the volume of water inside Cylinder A that lies below the bottom of Cylinder B (as in Figure 8).

      This cylinder has radius 12 and height 25h, and so has volume π(12)2(25h).

      Assume that when this volume of water has spilled into Cylinder B, it fills Cylinder B to a depth of d.

      Once Cylinder B is lowered to the bottom of Cylinder A, the volume of water in Cylinder B, π(9)2(d), must equal π(12)2(25h).

      Solving for d, we get 81πd=144π(25h) or d=3600144h81, and so d=40016h9.

      The depth of water in Cylinder B must be less than the height of Cylinder B (Cylinder B cannot be full), so then d<h or 40016h9<h or 40016h<9h or 400<25h, and so 16<h.

      As noted earlier, no water can spill into Cylinder B unless its height is less than that of Cylinder A, and so h<25.

      When Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full when 16<h<25.

      Solution 2

      Let the volume of Cylinder A be VA, the volume of Cylinder B be VB, and the initial volume of water be VW.

      As we determined in Solution 1, VA=3600π, VB=81πh, and VW=2736π.

      If VW+VB>VA, then water spills out of the large cylinder onto the ground.

      This gives 2736π+81πh>3600π or 81πh>864π, and so h>323.

      If 323<h<25, water will spill onto the ground and then into B. (If h323, B will actually be full of water when lowered into A, since no water spills out of A and the height of B is less than the initial height of water.)

      Assume that 323<h<25.

      Then the volume of water that spills out of Cylinder A onto the ground is Vwater on ground=VW+VBVA=81πh864π.

      When the tops of the two cylinders are at the same level (Figure 9), no water has spilled into Cylinder B, and so the volume of water in Cylinder A is the initial volume of water less the volume of water that has spilled out onto the ground.

      Figure 9

      That is, Vwater in A=2736πVwater on ground=2736π(81πh864π)=3600π81πh.

      From this point on, all water stays in Cylinder B or in Cylinder A.

      When Cylinder B is on the bottom of Cylinder A (Figure 10), the volume of water outside of Cylinder B (but inside Cylinder A), is the volume of Cylinder A that lies below the top of Cylinder B less the volume of Cylinder B.

      Figure 10

      That is, Vwater outside of B=π(122)hπ(92)h=63πh. Further, the volume of water in Cylinder A, 3600π81πh, must be equal to the volume of water outside of Cylinder B plus the volume of water inside of Cylinder B.

      That is, Vwater in A=Vwater outside of B+Vwater in BVwater in B=Vwater in AVwater outside of B=3600π81πh63πh=3600π144πh. The volume of water in Cylinder B must be less than the volume of Cylinder B, and so

      3600π144πh<81πh or 3600π<225πh, and thus 16<h.

      When Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full when 16<h<25.

    1. As a sum of one of more consecutive positive integers, 45 can be written as 45, 22+23, 14+15+16, 7+8+9+10+11, 5+6+7+8+9+10, and 1+2+3+4+5+6+7+8+9, and there are no other such lists.

      The value of C(45) is 6.

    2. The sum of the positive integers from 1 to n is given by the formula 12n(n+1).

      The sum of the positive integers from 4 to n (n4) is equal to the sum of the positive integers from 1 to n less the sum of the positive integers from 1 to 3, or 1+2+3=6.

      Therefore, m=4+5+6++n=12n(n+1)6=12(n(n+1)12)=12(n2+n12)=12(n3)(n+4). Since m=12(n+a)(n+b) with a<b, then a=3 and b=4.

    3. If m=(a+1)+(a+2)++n, for integers a0 and na+1, then m is equal to the sum of the integers from 1 to n less the sum of the integers from 1 to a.

      That is, m=12n(n+1)12a(a+1).

      Simplifying, we get m=12n(n+1)12a(a+1)=12(n2+na2a)=12(n2a2+na)=12((na)(n+a)+na)=12(na)(n+a+1) Each pair of integers (a,n) (a0 and na+1) for which m=12(na)(n+a+1) gives a unique sum of one or more consecutive positive integers from a+1 to n whose sum is m.

      In this question, we are asked to determine the number of such pairs (a,n) given that m=2×34×56.

      Since m=12(na)(n+a+1), then 2m=(na)(n+a+1).

      That is, 2m can be expressed as the product of two positive integers n+a+1 and na.

      The difference between these two integers is (n+a+1)(na)=2a+1, which is an odd integer for all integers a0.

      Since the difference between n+a+1 and na is odd, then one of these integers must be even and the other odd (we say that they have different parity).

      Thus, the problem of evaluating C(m) appears to be equivalent to counting the number of factor pairs of 2m (n+a+1 and na) that have different parity.

      At this point, we have shown that each pair of integers (a,n) (a0 and na+1) for which m=12(na)(n+a+1) gives a factor pair with different parity.

      We must now show that the converse is also true; that is, each factor pair with different parity gives a unique pair (a,n).

      Suppose that 2m=de for some positive odd integer d and positive even integer e.

      We show that each pair d and e will give a pair of integers a and n.

      If d>e, suppose that d=n+a+1 and e=na (since n+a+1>na).

      Adding the equations n+a+1=d and na=e, we get 2n+1=d+e or n=12(d+e1).

      Subtracting the two equations n+a+1=d and na=e, we get 2a+1=de or a=12(de1).

      Since d and e have different parity, then each of d+e and de is odd, and so each of d+e1 and de1 is even.

      Therefore, each of n=12(d+e1) and a=12(de1) is an integer and n>a.

      (If we assume that d<e, we can make a similar argument to show there exist corresponding integers a and n with n>a.)

      That is, each factor pair (d,e) having different parity gives a unique pair (a,n) with n>a.

      This confirms that evaluating C(m) is equivalent to counting the number of factor pairs of 2m that have different parity.

      Before evaluating C(2×34×56), we apply this to part (a) to confirm that C(45)=6, and to demonstrate that for each odd factor of 2×45, there exists a corresponding unique list of consecutive positive integers whose sum is 45.

      Since m=45=32×5, then 2m=2×32×5, and so the odd factors of 2×32×5 must be of the form 3i×5j for integers 0i2 and 0j1 (that is, odd numbers have only odd divisors).

      Since there are 3 choices for i (0,1,2), and 2 choices for j (0,1), there are 3×2=6 odd factors of 2×32×5 (these are 1,3,5,9,15, and 45).

      Next, we demonstrate that each of these 6 odd factors gives a unique pair (a,n) for which:

      1. (na)(n+a+1)=2×45, and

      2. n+a1 and na have different parity, and

      3. (a+1)+(a+2)++n=45.

      The odd factors 1,3,5,9,15,45 give the factor pairs (1,90),(3,30),(5,18),(9,10),(15,6), and (45,2) (we notice that the two numbers in each pair do indeed have different parity).

      Next we note that since a0, then n+a+1>na and so for example, using the factor pair (5,18), we get na=5 and n+a+1=18.

      Adding the two equations to solve this system of equations, we get 2n+1=23 and so n=11 and a=6.

      This pair (6,11) gives the sum 7+8+9+10+11=45.

      We summarize the results using the other factor pairs in the table below.

      Factor Pair an a+n+1 n a (a+1)+(a+2)++n
      (1,90) 1 90 45 44 45
      (3,30) 3 30 16 13 14+15+16=45
      (5,18) 5 18 11 6 7+8+9+10+11=45
      (9,10) 9 10 9 0 1+2+3+4+5+6+7+8+9=45
      (15,6) 6 15 10 4 5+6+7+8+9+10=45
      (45,2) 2 45 23 21 22+23=45

      Comparing this table to our answer in part (a), we see that indeed each odd factor of 2×45 gives a unique list of consecutive positive integers whose sum is 45.

      Finally, we turn our focus to evaluating C(2×34×56), that is, counting the number of odd factors of 22×34×56.

      The odd factors of 22×34×56 are of the form 3i×5j for integers 0i4 and 0j6.

      Since there are 5 choices for i and 7 choices for j, there are 5×7=35 odd factors of 22×34×56, and so C(2×34×56)=35.

    4. We would like to determine the smallest positive integer k for which C(k)=215=5×43 (both 5 and 43 are prime numbers).

      If k=2a for some non-negative integer a, then C(k)=1, and so k must have some odd prime factors p1,p2,,pn. Can you see why?

      That is, k=2ap1q1p2q2pnqn for distinct prime numbers pi,1in and positive integers qj,1jn.

      From part (c), we know that C(k)=(q1+1)(q2+1)(qn+1).

      Since C(k)=5×43=(q1+1)(q2+1)(qn+1), then we let n=2 and q1+1=5 or q1=4, and q2+1=43 or q2=42.

      To minimize k, we let a=0, and choose the smallest distinct odd primes p1=5 and p2=3 (p1=3 and p2=5 gives k=34×542, which is a much larger value for k).

      The smallest positive integer k for which C(k)=215 is k=54×342.