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2015 Canadian Intermediate
Mathematics Contest
Solutions

Wednesday, November 25, 2015
(in North America and South America)

Thursday, November 26, 2015
(outside of North American and South America)

©2015 University of Waterloo


PART A

  1. If 824=4x+3, then 8(x+3)=24(4) or 8x+24=96. Thus, 8x=72 and so x=9.
    Alternatively, we could note that 824=412 and so we have 824=412=4x+3 which gives x+3=12 or x=9

    Answer: 9

  2. We start by considering the ones (units) column of the given sum.
    From the units column, we see that the units digits of 3C must equal 6.
    The only digit for which this is possible is C=2. (We can check that no other digit works.)
    Thus, the sum becomes B2AB2+AB2876 We note that there is no “carry” from the ones column to the tens column.
    Next, we consider the tens column.
    From the tens column, we see that the units digit of 3B must equal 7.
    The only digit for which this is possible is B=9.
    Thus, the sum becomes 92A92+A92876 The “carry” from the tens column to the hundreds column is 2.
    Next, we consider the hundreds column.
    We see that the units digit of 2A+2 (the two digits plus the carry) must equal 8.
    Thus, the units digit of 2A must equal 6. This means that A=3 or A=8.
    The value of A=8 is too large as this makes A92=892 which is larger than the sum.
    Therefore, A=3.
    (Putting this analysis another way, we have 92+(100A+92)+(100A+92)=876. Simplifying, we obtain 200A+276=876 or 200A=600 and so A=3.)
    We can check that 92+392+392=876.) Therefore, A+B+C=3+9+2=14.

    Answer: 14

  3. Since the roof measures 5 m by 5 m, then its area is 52=25 m2.
    When this roof receives 6 mm (or 0.006 m) of rain, the total volume of rain that the roof receives is 250.006=0.15 m3. (We can imagine that the rain that falls forms a rectangular prism with base that is 5 m by 5 m and height 6 mm.)
    The rain barrel has diameter 0.5 m (and so radius 0.25 m) and a height of 1 m, and so has volume π0.2521=0.0625π m3.
    Thus, the percentage of the barrel that is full is 0.150.0625π×100%76.39% To the nearest tenth of a percent, the rain barrel will be 76.4% full of water.

    Answer: 76.4%

  4. Using exponent laws, we obtain the following equivalent equations: (24x23x)2=2x12242(x23x)=2x12242x26x=2x122(22)2x26x=2x12222(2x26x)=2x12224x212x2x1=122+4x212x(x1)=124x213x+3=1 Since 20=1, this last equation is true exactly when 4x213x+3=0 or (4x1)(x3)=0.
    Therefore, x=14 or x=3.
    We can check by substitution that each of these values of x satisfies the original equation.

    Answer: 14,3

  5. Suppose that Anna guesses “cat” c times and guesses “dog” d times.
    When she guesses “dog”, she is correct 95% of the time.
    When she guesses “cat”, she is correct 90% of the time.
    Thus, when she guesses “cat”, she is shown 0.9c images of cats and so c0.9c=0.1c images of dogs.
    Thus, when she guesses “dog”, she is shown 0.95d images of dogs and so d0.95d=0.05d images of cats.
    (We assume that c and d have the property that 0.9c and 0.95d are integers.)
    Therefore, the total number of images of cats that she is shown is 0.9c+0.05d and the total number of images of dogs that she is shown is 0.1c+0.95d.
    But the number of images of cats equals the number of images of dogs.
    Thus, 0.9c+0.05d=0.1c+0.95d, which gives 0.8c=0.9d or dc=0.80.9.
    Therefore, the ratio of the number of times that she guessed “dog” to the the number of times that she guessed “cat” is 8:9.

    Answer: 8:9

  6. Since X+Y=45, then tan(X+Y)=tan45=1.

    Now, tan(X+Y)=tanX+tanY1tanXtanY=1m+an11man=mn(1m+an)mn(11man)=n+ammna
    Thus, we have n+ammna=1.
    We want to determine the number of positive integers a for which this equation has exactly 6 pairs of positive integers (m,n) that are solutions.
    Re-arranging the equation, we obtain n+ammna=1n+am=mnaa=mnamna+a=mnamn+a2a=m(na)(na)2a=(m1)(na) So we want to determine the number of positive integers a for which (m1)(na)=2a has exactly 6 pairs of positive integers (m,n) that are solutions.
    Since a0, then m10 and so m1.
    Since m is a positive integer and m1, then m2; in other words, m1>0.
    Since m1>0 and 2a>0 and 2a=(m1)(na), then na>0 or n>a.
    Now, the factorizations of 2a as a product of two positive integers correspond with the pairs (m,n) of positive integers that are solutions to (m1)(na)=2a.
    This is because a factorization 2a=rs gives a solution m1=r (or m=r+1) and na=s (or n=s+a) and vice-versa.
    So the problem is equivalent to determining the number of positive integers a with a50 for which 2a has three factorizations as a product of two positive integers, or equivalently for which 2a has six positive divisors.
    Given an integer d and its prime factorization d=p1c1p2c2pkck, where p1,p2,,pk are distinct prime numbers and c1,c2,,ck are positive integers, the number of positive divisors of d is (c1+1)(c2+1)(ck+1).
    In order for this product to equal 6, d must have the form p5 for some prime number p or p2q for some prime numbers p and q. This is because the only way for 6 to be written as the product of two integers each larger than one is 23.
    In other words, we want 2a to be the fifth power of a prime, or the square of a prime times another prime.
    If 2a is of the form p5, then p=2 since 2a is already divisible by 2. Thus, 2a=25=32 and so a=16.
    If 2a is of the form p2q, then p=2 or q=2 since 2a is already divisible by 2.
    If p=2, then 2a=4q. Since a50, then 2a100 or q25, and so q can equal 3,5,7,11,13,17,19,23, giving a=6,10,14,22,26,34,38,46. (Note that q2.)
    If q=2, then 2a=2p2. Since 2a100, then p250 and so the prime p can equal 3,5,7 giving a=9,25,49. (Note that p2.)
    Therefore, in summary, there are 12 such values of a.

    Answer: 12

PART B

    1. The y-intercept of the line with equation y=2x+4 is 4.
      Since R is the point where this line crosses the y-axis, then R has coordinates (0,4).
      Since O has coordinates (0,0), the length of OR is 4.

    2. The point Q is the point of intersection of the line with equation y=2x+4 and the vertical line with equation x=p.
      Thus, the x-coordinate of Q is p, and its y-coordinate is thus 2p+4.
      So the coordinates of Q are (p,2p+4).

    3. When p=8, the coordinates of P are (8,0) and the coordinates of Q are (8,20).
      Quadrilateral OPQR is a trapezoid with parallel sides OR and PQ, and height equal to the length of OP, since OP is perpendicular to OR and PQ.
      Therefore, the area of OPQR is 12(OR+PQ)OP=12(4+20)8=96.

    4. In general, quadrilateral OPQR is a trapezoid with parallel sides OR and PQ, and height equal to the length of OP, since OP is perpendicular to OR and PQ.
      In terms of p, the area of OPQR is 12(OR+PQ)OP=12(4+(2p+4))p=12(2p+8)p=p(p+4) If the area of OPQR is 77, then p(p+4)=77 and so p2+4p77=0 or (p+11)(p7)=0.
      Since p>0, then p=7.
      We can verify that if p=7, then the area of OPQR is indeed 77.

    1. If f(r)=r, then rr1=r.
      Since r1, then r=r(r1) or r=r2r or 0=r22r and so 0=r(r2).
      Thus, r=0 or r=2.
      We can check by substitution that each of these values of r satisfies the original equation.

    2. Since f(x)=xx1, then f(f(x))=xx1xx11=xx(x1)=x1=x as required.
      (Note that above we have multiplied both the numerator and denominator of the complicated fraction by x1, which we can do since x1.)

    3. When xk, g(x)=2xx+k is well-defined.
      When xk and g(x)k, g(g(x))=2(2xx+k)2xx+k+k is well-defined.
      The following equations are equivalent for all x with xk and g(x)k: g(g(x))=x2(2xx+k)2xx+k+k=x4x2x+k(x+k)=x(multiplying numerator and denominator by x+k)4x=x(2x+k(x+k))4x=x(2x+kx+k2)4x=(k+2)x2+k2x0=(k+2)x2+(k24)x0=(k+2)(x2+(k2)x) Since there are values of x for which x2+(k2)x0, then to have (k+2)(x2+(k2)x)=0 for all x, we need to have k+2=0, or k=2. (For example, if k=1, the equation would become 3(x2x)=0 which only has the two solutions x=0 and x=1.)
      Therefore, g(g(x))=x for all x with xk and g(x)k exactly when k=2.

    4. When xcb, h(x)=ax+bbx+c is well-defined.
      When xcb and h(x)=ax+bbx+ccb, h(h(x))=a(ax+bbx+c)+bb(ax+bbx+c)+c is well-defined.
      The following equations are equivalent for all x with xcb and h(x)cb: h(h(x))=xa(ax+bbx+c)+bb(ax+bbx+c)+c=xa(ax+b)+b(bx+c)b(ax+b)+c(bx+c)=x(multiplying numerator and denominator by bx+c)a(ax+b)+b(bx+c)=x(b(ax+b)+c(bx+c))(a2+b2)x+(ab+bc)=(ab+bc)x2+(b2+c2)x0=b(a+c)x2+(c2a2)xb(a+c) Since this “quadratic equation” is satisfied for infinitely many real numbers x, then it must be the case that all three of its coefficients are 0.
      In other words, we must have b(a+c)=0 and c2a2=0, and b(a+c)=0.
      Since b0, then b(a+c)=0 gives a+c=0 or c=a.
      Note that if c=a, then c2a2=0 and b(a+c)=0 as well.
      Therefore, h(h(x))=x for all triples (a,b,c) of the form (a,b,a) where a and b are non-zero real numbers.

    1. When an=n2, we obtain

      n 1 2 3 4 5 6 7 8 9 10
      an 1 4 9 16 25 36 49 64 81 100
      bn 1
      Since b1a2, then b2=b1+a2=1+4=5.
      Since b2a3, then b3=b2+a3=5+9=14.
      Since b3a4, then b4=b3+a4=14+16=30.
      Since b4>a5, then b5=b4a5=3025=5.
      Since b5a6, then b6=b5+a6=5+36=41.
      Since b6a7, then b7=b6+a7=41+49=90.
      Since b7>a8, then b8=b7a8=9064=26.
      Since b8a9, then b9=b8+a9=26+81=107.
      Since b9>a10, then b10=b9a10=107100=7.

      In tabular form, we have

      n 1 2 3 4 5 6 7 8 9 10
      an 1 4 9 16 25 36 49 64 81 100
      bn 1 5 14 30 5 41 90 26 107 7

    2. As in (a), we start by calculating the first several values of bn:

      n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
      an 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
      bn 1 3 6 2 7 1 8 16 7 17 6 18 5 19 4 20 3 21 2 22 1 23

      We will show that if k is a positive integer with bk=1, then b3k+3=1 and bn1 for each n with k<n<3k+3.
      We note that this is consistent with the table shown above.
      Using this fact without proof, we see that b1=1, b6=1, b21=1, b66=1, b201=1, b606=1, b1821=1, b5466=1, and no other bn with n<5466 is equal to 1.
      (This list of values of bn comes from the fact that 6=3(1)+3, 21=3(6)+3, 66=3(21)+3, and so on.)
      Thus, once we have proven this fact, the positive integers n with n<2015 and bn=1 are n=1,6,21,66,201,606,1821.
      Suppose that bk=1.
      Since ak+1=k+1>1=bk, then bk+1=bk+ak+1=k+2.
      Since ak+2=k+2=bk+1, then bk+2=bk+1+ak+2=2k+4.
      Continuing in this way, we find that bk+2m1=k+3m for m=1,2,,k+2 and that bk+2m=2k+m+3 for m=1,2,,k+1.
      To justify these statements, we note that each is true when m=1 and that

      • if bk+2m=2k+m+3 for some m with 1mk+1, then since ak+2m+1=k+2m+1 and bk+2mak+2m+1=km+2>0 when mk+1, then bk+2m+1=bk+2mak+2m+1=km+2 which can be re-written as bk+2(m+1)1=k+3(m+1) and so is of the desired form; and

      • if bk+2m1=k+3m for some with 1mk+1, then since ak+2m=k+2m and bk+2m1ak+2m=33m0 when m1, then bk+2m=bk+2m1+ak+2m=(k+3m)+(k+2m)=2k+m+3 and so is of the desired form.

      This tells us that, for these values of m, the terms in the sequence bn have the desired form.
      Now bk+2m=2k+m+31 for m=1,2,,k+1 and bk+2m1=k+3m=1 only when m=k+2.
      Since k+2(k+2)1=3k+3, then b3k+3=1 and no other n with k<n<3k+3 gives bn=1.
      Thus, the positive integers n with n<2015 and bn=1 are n=1,6,21,66,201,606,1821.

    3. Suppose that a1,a2,a3, is eventually periodic.
      Then there exist positive integers r and p such that an+p=an for all nr.
      Note that this means that the terms from ar to ar+p1 repeat to give the terms from ar+p to ar+2p1, which repeat to give the terms from ar+2p to ar+3p1, and so on.
      In other words, the sequence a1,a2,a3, begins with r1 terms that do not necessarily repeat, and then continues with p terms (ar to ar+p1) which repeat indefinitely.
      Let M be the maximum value of the terms a1,a2,,ar1,ar,ar+1,,ar+p1.
      Since the terms in the sequence a1,a2,a3, are all positive integers, then M1.
      From the above comment, M is thus the maximum value of all of the terms in the sequence a1,a2,a3,.
      In other words, anM for all positive integers n.
      Next we prove that bn2M for all positive integers n.
      To do this, we first note that b1=a1M<2M.
      Next, we consider a given term bk in the sequence and assume that it satisfies bk2M.
      Note that bk+1 equals either bk+ak+1 or bkak+1.
      If bkM, then bk+1 is at most bk+ak+1.
      Since bkM and ak+1M, then bk+1M+M=2M.
      If M<bk2M, then since ak+1M, we obtain bk+1=bkak+1<bk2M.
      In either case, if bk2M, then bk+12M.
      Therefore, bn2M for all positive integers n.
      Now we consider the 2M pairs of terms (ar,br),(ar+p,br+p),,(ar+2Mp,br+2Mp).
      Since ar=ar+p=ar+2p=ar+3p=, then the first term in each of these pairs are equal.
      Since bn is a positive integer for each positive integer n and since bm2M, then, by the Pigeonhole Principle, at least two of the terms br,br+p,br+2p,,br+2Mp must be equal.
      (We have a list of 2M+1 numbers each of which has at most 2M possible values, so two of the numbers must be the same.)
      Suppose that br+sp=br+tp for some non-negative integers s<t.
      Now each term bk+1 is completely determined by the numbers bk and ak+1. (If bk1=bk2 and ak1+1=ak2+1, then bk1+1=bk2+1.)
      Since ar+sp+1=ar+tp+1 and ar+sp+2=ar+tp+2 and ar+sp+3=ar+tp+3 and so on, then br+sp+1=br+tp+1 and br+sp+2=br+tp+2 and so on. (Since the terms in the two sequences are equal at the same two points, then all subsequent terms will be equal.)
      In particular, if br+sp=br+tp>ar+sp+1=ar+tp+1, then br+sp+1=br+spar+sp+1=br+tpar+tp+1=br+tp+1 and if br+sp=br+tpar+sp+1=ar+tp+1, then br+sp+1=br+sp+ar+sp+1=br+tp+ar+tp+1=br+tp+1 This argument then repeats at each step going forward.
      In other words, starting at each of br+sp and br+tp, the terms going forward will be equal, which means that the sequence b1,b2,b3, is eventually periodic. (We note that if R=r+sp and P=(ts)p, then bN=bN+P for each positive integer NR.)