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2020 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 18, 2020
(in North America and South America)

Thursday, November 19, 2020
(outside of North American and South America)

©2020 University of Waterloo


Part A

  1. Between them, Markus and Katharina have 9+5=14 candies.
    When Sanjiv gives 10 candies in total to Markus and Katharina, they now have 14+10=24 candies in total.
    Since Markus and Katharina have the same number of candies, they each have 1224=12 candies.

    Answer: 12

  2. Suppose that the square has side length 2x cm.
    Each of the two rectangles thus has width 2x cm and height x cm.
    In terms of x, the perimeter of one of these rectangles is 2(2x cm)+2(x cm) which equals 6x cm.
    Since the perimeter of each rectangle is 24 cm, then 6x=24 which means that x=4.
    Since the square has side length 2x cm, then the square is 8 cm by 8 cm and so its area is 64 cm2.

    Answer: 64 cm2

  3. Solution 1:

    Since a, b, c, d, and e are consecutive with a<b<c<d<e, we can write b=a+1 and c=a+2 and d=a+3 and e=a+4.
    From a2+b2+c2=d2+e2, we obtain the equivalent equations: a2+(a+1)2+(a+2)2=(a+3)2+(a+4)2a2+a2+2a+1+a2+4a+4=a2+6a+9+a2+8a+16a28a20=0(a10)(a+2)=0 Since a is positive, then a=10.
    (Checking, we see that 102+112+122=100+121+144=365 and 132+142=169+196=365.)

    Solution 2:

    Since a, b, c, d, and e are consecutive with a<b<c<d<e, we can write b=c1 and a=c2 and d=c+1 and e=c+2.
    From a2+b2+c2=d2+e2, we obtain the equivalent equations: (c2)2+(c1)2+c2=(c+1)2+(c+2)2c24c+4+c22c+1+c2=c2+2c+1+c2+4c+4c212c=0c(c12)=0 Since c is a positive integer, then c=12, which means that a=c2=10.
    (Checking, we see that 102+112+122=100+121+144=365 and 132+142=169+196=365.)

    Answer: a=10

  4. We note that π3.14159 which means that 3.14<π<3.15.
    Therefore, π+0.85 satisfies 3.99<π+0.85<4.00 and π+0.86 satisfies 4.00<π+0.86<4.01.
    This means that 3<π<π+0.01<π+0.02<<π+0.85<4<π+0.86<π+0.87<<π+0.99<5 Also, π+0.85=3 because π+0.85 is between 3 and 4, and π+0.86=4 because π+0.86 is between 4 and 5.
    Next, we re-write S=π+π+1100+π+2100+π+3100++π+99100 as S=π+0.00+π+0.01+π+0.02+π+0.03++π+0.84+π+0.85+π+0.86+π+0.87++π+0.99 Each of the terms π+0.00, π+0.01, π+0.02, π+0.03, , π+0.84, π+0.85 is equal to 3, since each of π+0.00, π+0.01, , π+0.85 is greater than 3 and less than 4.
    Each of the terms π+0.86, π+0.87, , π+0.99 is equal to 4, since each of π+0.86, π+0.87, , π+0.99 is greater than 4 and less than 5.
    There are 86 terms in the first list and 14 terms in the second list.
    Thus, S=863+144=863+143+14=1003+14=314.

    Answer: S=314

  5. We square the two given equations to obtain (3sinx+4cosy)2=52(4siny+3cosx)2=22 or 9sin2x+24sinxcosy+16cos2y=2516sin2y+24sinycosx+9cos2x=4 Adding these equations and re-arranging, we obtain 9sin2x+9cos2x+16sin2y+16cos2y+24sinxcosy+24cosxsiny=29 Since sin2θ+cos2θ=1 for every angle θ, then 9+16+24(sinxcosy+cosxsiny)=29 from which we obtain sinxcosy+cosxsiny=424 and so sin(x+y)=16. (This uses the Useful Fact for Part A.)
    (It is possible to solve for sinx, cosx, siny, and cosy. Can you see an approach that would allow you to do this?)

    Answer: 16

  6. Using the second property with x=0, we obtain f(0)=12f(0) from which we get 2f(0)=f(0) and so f(0)=0.
    Using the first property with x=0, we obtain f(1)=1f(0)=10=1.
    Using the first property with x=12, we obtain f(12)=1f(12) and so 2f(12)=1 which gives f(12)=12.
    Using the second property with x=1, we obtain f(13)=12f(1)=12.
    Next, we note that 370.43.
    Since 3712, then, using the third property, f(37)f(12)=12.
    Since 3713, then, using the third property, f(37)f(13)=12.
    Since 12f(37)12, then f(37)=12.
    Using the second property with x=37, we obtain f(17)=12f(37)=1212=14.
    Using the first property with x=17, we obtain f(67)=1f(17)=114=34.
    Here are two additional comments about this problem and its solution:

    1. While the solution does not contain many steps, it is not easy to come up with the best steps in the best order to actually solve this problem.

    2. There is indeed at least one function, called the Cantor function, that satisfies these properties. This function is not easy to write down, and finding the function is not necessary to answer the given question. For those interested in learning more, consider investigating ternary expansions and binary expansions of real numbers between 0 and 1, as well as something called the Cantor set.

    Answer: f(67)=34

Part B

    1. To determine the point of intersection, we equate the two expressions for y to successively obtain: 4x32=6x+810x=40x=4 When x=4, using the equation y=4x32, we obtain y=4432=16.
      Therefore, the lines intersect at (4,16).

    2. To determine the point of intersection, we equate the two expressions for y to successively obtain: x+3=2x3a23+3a2=3xx=1+a2 When x=1+a2, using the equation y=x+3, we obtain y=(1+a2)+3=2a2.
      Therefore, the lines intersect at (1+a2,2a2).

    3. Since c is an integer, then c2 is a integer that is less than or equal to 0.
      The two lines have slopes c2 and 1. Since c20, these slopes are different, which means that the lines are not parallel, which means that they intersect.
      To determine the point of intersection, we equate the two expressions for y to successively obtain: c2x+3=x3c23+3c2=x+c2x3+3c2=x(1+c2) Since c20, then 1+c21, which means that we can divide both sides by 1+c2 to obtain x=3+3c21+c2=3.
      In particular, this means that the x-coordinate of the point of intersection is an integer.
      When x=3, using the equation y=c2x+3, we obtain y=c23+3=33c2.
      Since c is an integer, then y=33c2 is an integer.
      Therefore, the lines intersect at a point whose coordinates are integers.

    4. To determine the point of intersection in terms of d, we equate the two expressions for y to successively obtain: dx+4=2dx+22=dx For the value of x to be an integer, we need d0 and 2d to be an integer.
      Since d is itself an integer, then d is a divisor of 2, which means that d equals one of 1, 1, 2, 2.
      We still need to confirm that, for each of these values of d, the coordinate y is also an integer.
      When x=2d, using the equation y=dx+4, we obtain y=d(2d)+4=2+4=6.
      Therefore, when d=1,1,2,2, the lines intersect at a point with integer coordinates.
      We can verify this in each case:

      • d=1: The lines with equations y=x+4 and y=2x+2 intersect at (2,6).

      • d=1: The lines with equations y=x+4 and y=2x+2 intersect at (2,6).

      • d=2: The lines with equations y=2x+4 and y=4x+2 intersect at (1,6).

      • d=2: The lines with equations y=2x+4 and y=4x+2 intersect at (1,6).

    1. Each interior angle in a regular hexagon measures 120. (One way to verify this is to use the fact that the sum of the interior angles in a regular polygon with n sides is (n2)180. When n=6, this sum is 4180=720. In a regular hexagon, each of these angles has measure 16720=120.)
      Since 120 is equal to 13 of 360, then the area of the shaded sector is 13 of the area of a complete circle of radius 6.
      Therefore, the shaded area is 13π(62)=12π.

    2. To determine the area of region between the arc through C and E and the line segment CE, we take the area of the sector obtained in (a) and subtract the area of CDE.
      CDE has DE=DC=6 and CDE=120.
      There are many ways to find the area of this triangle.
      One way is to consider ED as the base of this triangle and to draw an altitude from C to ED extended, meeting ED extended at T.

      Two regions in the interior of hexagon ABCDEF are shaded. One region is enclosed by circular arc C E and line segment C E. The other region is enclosed by circular arc B F and line segment B F.

      Since CDE=120, then CDT=180CDE=60.
      This means that CDT is a 30-60-90 triangle, and so CT=32CD=326=33.
      This means that the area of CDE is 12EDCT=12633=93.
      Therefore, the area of the shaded region between the arc through C and E and the line segment CE is 12π93.
      Since the region between line segment BF and the arc through B and F is constructed in exactly the same way, its area is the same.
      Therefore, the total area of the shaded regions is 2(12π93)=24π183.

    3. Let M be the midpoint of DE and N be the midpoint of EF.
      This means that M and N are the centres of two of the semi-circles.
      Let P be the point other than E where the semi-circles with centres M and N intersect.
      Join E to P.

      Six regions in the interior of hexagon ABCDEF are shaded. Each region lies inside exactly two semi-circles whose diameters are edges of the hexagon. One shaded region is enclosed by circular arc E D with centre M and circular arc F E with centre N. This region is divided in two by line segment E P.

      By symmetry, EP divides the shaded region between these two semi-circles into two pieces of equal area.
      Let the area of one of these pieces be a.
      Furthermore, by symmetry in the whole hexagon, each of the six shaded regions between two semi-circles is equal in area.
      This means that the entire shaded region is equal to 12a.
      Therefore, we need to determine the value of a.
      Consider the region between EP and the arc with centre M through E and P.

      Since DE=EF=6, then DM=ME=EN=NF=3. Each of the two semi-circles has radius 3.
      Since P is on both semi-circles, then NP=MP=3.
      Consider EMP. Here, we have ME=MP=3.
      Also, MEP=60 since DEF=120 and MEP=NEP by symmetry.
      Since ME=MP, then MPE=MEP=60.
      Since EMP has two 60 angles, then its third angle also has measure 60, which means that EMP is equilateral.
      Therefore, PE=3 and EMP=60.
      Now, we can calculate the area a.
      The area a is equal to the area of the sector of the circle with centre M defined by E and P minus the area of EMP.
      Since EMP=60, which is 16 of a complete circle, and the radius of the circle from which the sector comes is 3, then the area of the sector is 16π(32)=32π.
      The area of EMP, which is equilateral with side length 3, can be found in many different ways.
      One way to do this is to use the formula that the area of a triangle with two side lengths x and y and an angle of θ between these two sides is equal to 12xysinθ.
      Thus, the area of EMP is 1233sin60=934.
      This means that a=32π934.
      Finally, the total area of the shaded regions equals 12a which equals 12(32π934), or 18π273.

    1. When p=33 and q=216, f(x)=x333x2+216x=x(x233x+216)=x(x9)(x24) since 9+24=33 and 924=216, and g(x)=3x266x+216=3(x222x+72)=3(x4)(x18) since 4+18=22 and 418=72.
      Therefore, the equation f(x)=0 has three distinct integer roots (namely x=0, x=9 and x=24) and the equation g(x)=0 has two distinct integer roots (namely x=4 and x=18).

    2. Suppose first that the equation f(x)=0 has three distinct integer roots.
      Since f(x)=x3px2+qx=x(x2px+q), then these roots are x=0 and the roots of the quadratic equation x2px+q=0 which are x=p±p24(1)q2(1)=p±p24q2 For the roots of x2px+q=0 to be distinct, we need p24q to be positive.
      For the roots of x2px+q=0 to be integers, we need each of p±p24q to be an integer, which means that p24q is an integer, which means that p24q must be a perfect square.
      Therefore, p24q is a positive perfect square.
      Suppose also that the equation g(x)=0 has two distinct integer roots.
      The roots of the equation 3x22px+q=0 are x=2p±(2p)24(3)(q)2(3)=2p±4p212q6=p±p23q3 As above, for these roots to be distinct, we need p23q to be positive and a perfect square.
      Furthermore, since the roots of the equation 3x22px+q=0 are distinct integers, then the roots of the equation x22p3x+q3=0 are also distinct integers.
      This means that 2p3 and q3, which are the sum and product of the roots, respectively, are themselves integers.
      This means that p must be a multiple of 3 and q must be a multiple of 3.
      To complete this part, we need to prove that q (which we know is a multiple of 3) is in fact a multiple of 9.
      To do this, we use the fact that p and q are multiples of 3 and that p24q is a perfect square.
      Since p and q are multiples of 3, we can set p=3P and q=3Q for some integers P and Q.
      In this case, p24q=(3P)24(3Q)=9P212Q=3(3P24Q) This means that p24q is a perfect square that is a multiple of 3.
      Since any perfect square that is a multiple of 3 must be a multiple of 9 (prime factors of perfect squares occur in pairs), then 3P24Q is itself a multiple of 3.
      Since 3P24Q is a multiple of 3 and 3P2 is a multiple of 3, then 4Q must be a multiple of 3, which means that Q is a multiple of 3.
      Since q=3Q and Q is a multiple of 3, then q is a multiple of 9, which completes this part.

    3. The goal of this solution is to show that there are infinitely many pairs of positive integers (p,q) with certain properties. To do this, we do not have to find all pairs (p,q) with these properties, as long as we still find infinitely many such pairs. This means that we can make some assumptions as we go. Rather than making all of these assumptions at the very beginning, we will add these as we go.
      To begin, we assume that p and q are positive integers with p a multiple of 3 and q a multiple of 9. (Assumption #1)
      Thus, we write p=3a and q=9b for some positive integers a and b.
      Suppose that a and b have the additional property that a23b=m2 and a24b=n2 for some positive integers m and n. (Assumption #2)
      These first two Assumptions are not surprising given the results of (b).
      In this case, the non-zero solutions of f(x)=0 are x=p±p24q2=3a±(3a)24(9b)2=3a±3a24b2=3a±3n2 and the solutions of g(x)=0 are x=2p±4p212q6=p±p23q3=3a±3a23b3=a±m These solutions are all integers as long as the integers 3a±3n are both even, which is equivalent to saying that a and n are both even or both odd (that is, have the same parity).
      Since a24b=n2, this means that a2n2=4b, which is even, which means that a2 and n2 have the same parity, which means that a and n have the same parity.
      Further, we note that since p=3a and q=9b then both p and q are divisible by 3 and so gcd(p,q)=3 exactly when a and 3b have no further common divisors larger than 1.
      Therefore, to find an infinite number of pairs of positive integers (p,q) which satisfy the given conditions, we can find an infinite number of pairs of positive integers (a,b) for which a23b and a24b are both positive perfect squares, and where gcd(a,3b)=1.
      Recall that a23b=m2 and a24b=n2 for some positive integers m and n.
      This gives 4a212b=4m2 and 3a212b=3n2.
      Subtracting, we obtain a2=4m23n2, which gives 3n2=4m2a2.
      We re-write this equation as n23=(2m+a)(2ma).
      Now we suppose that 2m+a=n22ma=3 (Assumption #3)
      This adds a further assumption that connects the integers a, b, m, and n, and allows us to start representing these variables in terms of just n.
      Then 4m=n2+3 which gives m=n2+34 and 2a=n23 which gives a=n232.
      Under these assumptions, for m and a to be integers, we need n to be odd.
      Recall that, to find an infinite number of pairs of positive integers (p,q) which satisfy the given conditions, we can find an infinite number of pairs of positive integers (a,b) for which a23b and a24b are both positive perfect squares, and where gcd(a,3b)=1.
      Setting n=2N+1 for some positive integer N, we obtain m=(2N+1)2+34=4N2+4N+1+34=N2+N+1a=n232=4N2+4N+132=2N2+2N1 We can use these now to write b=a2n24=(2N2+2N1)2(2N+1)24=(2N2+2N1+2N+1)(2N2+2N12N1)4=(N2+2N)(N21) We note that the desired relationships between a, b, m, and n still hold:

      • Because b=a2n24, we have a24b=n2.

      • Because 2m+a=n2 and 2ma=3, then 4m2a2=3n2.

      • This gives 4m2a2=3(a24b) and so 4m2=4a212b and so a23b=m2.

      Therefore, each positive integer N defines integers a and b with the property that a23b and a24b are both perfect squares.
      Therefore, in order to complete our proof, we need to show that there are infinitely many integers N for which gcd(a,3b)=1.
      Since a=2N2+2N1 and b=(N2+2N)(N21)=N(N+2)(N+1)(N1), we want to show that there are infinitely many integers N for which gcd(2N2+2N1,3N(N+2)(N+1)(N1))=1 We consider 2N2+2N1 and N(N+1)=N2+N.
      Since gcd(A,B)=gcd(A,BQA) for all integers A,B,Q, then gcd(N2+N,2N2+2N1)=gcd(N2+N,2N2+2N12(N2+N))=gcd(N2+N,1) The only positive divisor of 1 is 1, so gcd(N2+N,2N2+2N1)=gcd(N2+N,1)=1 Since gcd(2N2+2N1,N2+N)=1 and gcd(A,BC)=gcd(A,B) when gcd(A,C)=1, then gcd(2N2+2N1,3N(N+2)(N+1)(N1))=gcd(2N2+2N1,3(N+2)(N1)(N2+N))=gcd(2N2+2N1,3(N+2)(N1))=gcd(2N2+2N1,3N2+3N6) Since 2N2+2N1 is odd, then gcd(2N2+2N1,2)=1.
      Thus, gcd(2N2+2N1,3N2+3N6)=gcd(2N2+2N1,2(3N2+3N6)) Again using the fact that gcd(A,B)=gcd(A,BQA), we obtain gcd(2N2+2N1,6N2+6N12)=gcd(2N2+2N1,6N2+6N123(2N2+2N1))=gcd(2N2+2N1,9) Therefore, gcd(2N2+2N1,3N(N+2)(N+1)(N1))=gcd(2N2+2N1,9) This means that, to complete our proof, we need to show that there are infinitely many positive integers N for which gcd(2N2+2N1,9)=1.
      Note that the positive divisors of 9 are 1, 3 and 9.
      Suppose that N is a multiple of 3. In this case 2N2+2N is a multiple of 3 (because it is a multiple of N), which means that 2N2+2N1 is not a multiple of 3, which means that gcd(2N2+2N1,9)=1.
      Therefore, there are infinitely many positive integers N for which gcd(2N2+2N1,6N(N+2)(N+1)(N1))=gcd(2N2+2N1,9)=1 This means that there are infinitely many positive integers N for which gcd(a,3b)=1.
      This means that there are infinitely many pairs of positive integers (a,b) for which a23b and a24b are both positive perfect squares and where gcd(a,3b)=1.
      This means that there are infinitely many pairs of positive integers (p,q) with the required properties.