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2017 Fermat Contest
Solutions
(Grade 11)

Tuesday, February 28, 2017
(in North America and South America)

Wednesday, March 1, 2017
(outside of North American and South America)

©2016 University of Waterloo


  1. Evaluating, 6×20172017×4=2017(64)=2017(2)=4034.

    Answer: (D)

  2. In the diagram, there are 7 rows in which there are shaded squares, and there are 7 shaded squares in each row.
    Thus, there are 77=49 shaded squares.

    Answer: (E)

  3. The sum of 2, 3 and 6 is 2+3+6=11. Their product is 236=36.

    Answer: (C)

  4. Since 300 litres drains in 25 hours, then the rate at which water is leaving the tank equals 300 L25 h or 12 L/h.

    Answer: (A)

  5. The graph of y=2x2+4 is a parabola.
    Since the coefficient of x2 is negative, the parabola opens downwards.
    Since the constant term is positive, the y-intercept of the parabola (that is, the value of y when x=0) is positive.
    Of the given graphs, only (D) has these properties. (Since the coefficient of x is 0 in the given equation, then the graph should be symmetric about the y-axis, as the graph in (D) is.)

    Answer: (D)

  6. Since the average of 5 and 9 is 5+92=7, then the averages of 5 and x and of x and 9 must be 10 and 12.
    In other words, 5+x2 and x+92 are equal to 10 and 12 in some order.
    Adding these, we obtain 5+x2+x+92=10+12 or 14+2x2=22 and so 7+x=22 or x=15.
    (We could have also noted that 5+x2<x+92 since 5<9, and so x+92=12.)

    Answer: (B)

  7. Since x=1 is a solution of the equation x2+ax+1=0, then 12+a(1)+1=0 or 2+a=0 and so a=2.

    Answer: (E)

  8. Since 12n+14n=24n+14n=34n, then the given equation becomes 34n=312 or 4n=12.
    Thus, n=3.

    Answer: (E)

  9. We need to determine the time 100 hours before 5 p.m. Friday.
    Since there are 24 hours in 1 day and since 100=4(24)+4, then 100 hours is equal to 4 days plus 4 hours.
    Starting at 5 p.m. Friday, we move 4 hours back in time to 1 p.m. Friday and then an additional 4 days back in time to 1 p.m. Monday.
    Thus, Kamile turned her computer on at 1 p.m. Monday.

    Answer: (D)

  10. Suppose that the integers a<b<c<n have a+b+c+n=100.
    Since a<b<c<n, then a+b+c+n<n+n+n+n=4n. Thus, 100<4n and so n>25.
    Since n is an integer, then n is at least 26.
    Could n be 26? In this case, we would have a+b+c=10026=74.
    If n=26, then a+b+c is at most 23+24+25=72, which means that we cannot have a+b+c=74.
    Therefore, n cannot be 26.
    Could n be 27? In this case, we would have a+b+c=10027=73.
    Here, we could have a+b+c=23+24+26=73, and so n=27 is possible, which means that the smallest possible value of n is 27. (There are other values of a,b,c that work with n=27 as well.)

    Answer: (D)

  11. Each student brought exactly one of an apple, a banana, and an orange.
    Since 20% of the students brought an apple and 35% brought a banana, then the percentage of students who brought an orange is 100%20%35%=45%.
    Therefore, the 9 students who brought an orange represent 45% of the class.
    This means that 1 student represents 45%÷9=5% of the class.
    Thus, the class has 100%÷5%=20 students in it.

    Answer: (D)

  12. The question is equivalent to asking how many three-digit positive integers beginning with 2 are larger than 217.
    These integers are 218 through 299 inclusive.
    There are 299217=82 such integers.

    Answer: (B)

  13. The line through R(2,4) and Q(4,0) has slope 4024=2.
    Since it passes through (4,0), this line has equation y0=2(x4) or y=2x+8.
    The line with equation y=2x+8 has y-intercept 8, and so the coordinates of P are (0,8).
    Now, OPQ is right-angled at O and so its area is 12(OQ)(OP)=12(4)(8)=16.

    Answer: (E)

  14. The expression (1+12)(1+13)(1+14)(1+15)(1+16)(1+17)(1+18)(1+19) is equal to (32)(43)(54)(65)(76)(87)(98)(109) which equals 34567891023456789 Removing common factors from the numerator and denominator, we obtain 102 or 5.

    Answer: (A)

  15. Since XMZ=30, then XMY=180XMZ=18030=150.
    Since the angles in XMY add to 180, then YXM=180XYZXMY=18015150=15 (Alternatively, since XMZ is an exterior angle of XMY, then XMZ=YXM+XYM which also gives YXM=15.)
    Since XYM=YXM, then XMY is isosceles with MX=MY.
    But M is the midpoint of YZ, and so MY=MZ.
    Since MX=MY and MY=MZ, then MX=MZ.
    This means that XMZ is isosceles with XZM=ZXM.
    Therefore, XZY=XZM=12(180XMZ)=12(18030)=75.

    Answer: (A)

  16. Since x+2y=30, then x5+2y3+2y5+x3=x5+2y5+x3+2y3=15x+15(2y)+13x+13(2y)=15(x+2y)+13(x+2y)=15(30)+13(30)=6+10=16

    Answer: (B)

  17. Suppose that the base of the prism is b cm by w cm and the height of the prism is h cm.
    Since Aaron has 144 cubes with edge length 1 cm, then the volume of the prism is 144 cm3, and so bwh=144.
    Since the perimeter of the base is 20 cm, then 2b+2w=20 or b+w=10.
    Since b and w are positive integers, then we can make a chart of the possible combinations of b and w and the resulting values of h=144bw, noting that since b and w are symmetric, then we can assume that bw:

    b w h
    1 9 16
    2 8 9
    3 7 487
    4 6 6
    5 5 14425

    Since h must itself be a positive integer, then the possible values of h are 16, 9 and 6.
    The sum of the possible heights is 16 cm+9 cm+6 cm=31 cm.

    Answer: (A)

  18. For any positive real number x, x equals the largest integer less than or equal to x and so xx.
    In particular, xxxx=x2.
    Thus, if xx=36, then 36x2.
    Since x>0, then x6.
    In fact, if x=6, then x=6=6 and so xx=x2=36. Therefore, x=6. (Note that if x>6, then xx>66=36.)
    Also, since yy=71, then y271.
    Since y>0, then y718.43.
    Since y718.43, then y8.
    Suppose that y=8.
    In this case, y=71y=718=8.875. Note that if y=718, then y=8, so y=718 is a solution. (In fact, it is the only solution with y>0. Can you see why?)
    Therefore, x+y=6+718=1198.

    Answer: (B)

  19. If a>0, the distance from the vertical line with equation x=a to the y-axis is a.
    If a<0, the distance from the vertical line with equation x=a to the y-axis is a.
    In each case, there are exactly two points on the vertical line with equation x=a that are also a distance of a or a (as appropriate) from the x-axis: (a,a) and (a,a). These points lie on the horizontal lines with equations y=a and y=a, respectively.
    (If a=0, the line x=a coincides with the y-axis and the unique point on this line that is equidistant from the coordinate axes is the origin (0,0) which does not lie on the line with equation 3x+8y=24.)
    If the point (a,a) lies on the line 3x+8y=24, then 3a+8a=24 or a=2411.
    If the point (a,a) lies on the line 3x+8y=24, then 3a8a=24 or a=245.
    The sum of these values of a is 2411+(245)=12026455=14455.

    Answer: (B)

  20. Since m and n are positive integers with n>1 and mn=225×340, then 2 and 3 are prime factors of m (since they are prime factors of mn) and must be the only prime factors of m (since if there were other prime factors of m, then there would be other prime factors of mn).
    Therefore, m=2a×3b for some positive integers a and b and so mn=(2a×3b)n=2an×3bn.
    Since mn=225×340, then we must have an=25 and bn=40.
    Since a,b,n are positive integers, then n is a common divisor of 25 and 40.
    Since n>1, then n=5, which means that a=5 and b=8.
    In this case, m=25×38=32×6561=209952, which gives m+n=209952+5=209957.

    Answer: (C)

  21. Since WXYZ is a four-digit positive integer, then WXYZ9999. (In fact WXYZ cannot be this large since all of its digits must be different.)
    Since WXYZ9999, then TWUYV2(9999)=19998.
    Since T0, then T=1.
    Next, we note that the “carry" from any column to the next cannot be larger than 1. (Since Z9, then Z+Z18 and so the carry from the ones column to the tens column is 0 or 1. Similarly, since Y+Y18, then the largest sum of the digits plus carry in the tens column is 19 and so the maximum carry to the hundreds column is 1. This reasoning continues in the columns to the left.)
    Thus, we make a chart of possible digits d and the resulting units digit in the sum from d+d with and without a carry of 1:

    d Units digit of d+d with no carry Units digit of d+d with carry of 1
    0 0 1
    1 2 3
    2 4 5
    3 6 7
    4 8 9
    5 0 1
    6 2 3
    7 4 5
    8 6 7
    9 8 9

    We use this table to first determine the digits W and Y.
    Since the digits in the thousands column are all the same, then the digit W must be 9, since it must be at least 5 to produce a carry to the ten thousands column. We note further that this means that X5 to produce a carry into this column.
    Also, the digit Y must equal 0 (since the digits T,U,V,W,X,Y,Z are different).
    This means that there is no carry from the ones column to the tens column.
    We summarize what we know so far:

    Vertical addition of two equal numbers, the digits of which from left to right are 9 X 0 Z. They sum to 1 9 U 0 V.

    and X5 and Z4.
    Since T=1 and W=9, then Z can be 2, 3 or 4, and X can be 5, 6, 7, or 8.
    Note that if X=5, then we have U=0=Y, which is not possible, so X5.
    If Z=2, then V=4. In this case, we cannot have X=6 (which would give U=2=Z) or X=7 (which would give U=4=V) and so X=8, which gives U=6.
    If Z=3, then V=6. In this case, X cannot equal 6 or 8 and so X=7 (which gives U=4).
    If Z=4, then V=8. In this case, X cannot equal 7 or 8 and so X=6 (which gives U=2).
    In summary, there are 3 possible values for U, namely, 2, 4 and 6.
    We can check that the sums 9802+9802=19604 and 9703+9703=19406 and 9604+9604=19208 all satisfy the original problem.

    Answer: (C)

  22. We slice the cylinder, cone and sphere using a vertical plane that passes through the centres of the top and bottom faces of the cylinder and through the centre of the sphere.
    The resulting cross-sections of the cylinder, cone and sphere are a rectangle, triangle and circle, respectively.
    Since the sphere is touching the cylinder and the cone, then by slicing the cylinder in this way, the resulting circle is tangent to two sides of the rectangle (at F and H) and a side of the triangle (at G).
    Join O to F, G and H. Since radii are perpendicular to tangents at the resulting points of tangency, then OF is perpendicular to AD, OG is perpendicular to AE, and OH is perpendicular to DE.
       

    Let the radius of the sphere (now a circle) be r. Then OF=OG=OH=r.
    Since the radius of the cylinder is 12, then DE=12.
    Since the height of the cylinder is 30, then AD=30.
    Since FOHD has right angles at F, D and H, then it must have four right angles, and so is a rectangle.
    Since OF=OH=r, then FOHD is in fact a square with DH=DF=r.
    Since DE=12 and DH=r, then EH=12r.
    Since AD=30 and DF=r, then AF=30r.
    Since AG and AF are tangents to the circle from the same point, then AG=AF=30r. (To see this, note that AFO and AGO are both right-angled, have a common side AO and equal sides FO and GO, which means that they are congruent.)
    Similarly, EG=EH=12r.
    Finally, AE=AG+GE.
    By the Pythagorean Theorem in ADE, AE=122+302=1044.
    Thus, 1044=(30r)+(12r) and so 2r=421044 or r=211210444.8445.
    Of the given choices, this is closest to 4.84.

    Answer: (A)

  23. Since a is a positive integer and a+bc is a positive integer, then bc is a positive integer. In other words, b is a multiple of c.
    Similarly, since ac+b is a positive integer and b is a positive integer, then a is a multiple of c.
    Thus, we can write a=Ac and b=Bc for some positive integers A and B.
    Therefore, a+bc=101 becomes Ac+B=101 and ac+b=68 becomes A+Bc=68.
    Adding these new equations gives Ac+B+A+Bc=101+68 or A(c+1)+B(c+1)=169 and so (A+B)(c+1)=169.
    Since (A+B)(c+1)=169, then c+1 is a divisor of 169.
    Since 169=132, then the positive divisors of 169 are 1, 13, 169.
    Since A,B,c are positive integers, then A+B2 and c+12.
    Since neither A+B nor c+1 can equal 1, then A+B=c+1=13.
    Finally, a+bc=Ac+Bcc=A+B=13 and so k=13.

    Answer: (A)

  24. We label the 8 teams as F, G, H, J, K, L, M, N.
    We first determine the total number of games played.
    Since each pair of teams plays exactly one game, then each team plays 7 games (one against each of the other 7 teams). Since there are 8 teams, then it seems as if there are 87=56 games, except that each game has been counted twice in this total (since, for example, we have counted G playing K and K playing G). Thus, there are in fact 872=28 games played.
    Since there are 28 games played and there are 2 equally likely outcomes for each game, then there are 228 possible combinations of outcomes. (We can consider that the games are numbered from 1 to 28 and that F plays G in game 1, F plays H in game 2, and so on. A possible combination of outcomes for the tournament can be thought of as a “word” with 28 letters, the first letter being F or G (depending on the winner of the first game), the second letter being F or H (depending on the winner of the second game), and so on. There are two choices for each letter in the word, and so 228 possible words.)
    To determine the probability that every team loses at least one game and every team wins at least one game, we determine the probability that there is a team that loses 0 games or a team that wins 0 games and subtract this probability from 1.
    Since we know the total number of possible combinations of outcomes, we determine the probability by counting the number of combinations of outcomes in which there is a team that loses 0 games (that is, wins all of its games) or a team that wins 0 games (that is, loses all of its games), or both.
    To determine the number of combinations of outcomes in which there is a team that wins all of its games, we note that there are 8 ways to choose this team. Once a team is chosen (we call this team X), the results of the 7 games played by X are determined (X wins all of these) and the outcomes of the remaining 287=21 games are undetermined.
    Since there are two possible outcomes for each of these 21 undetermined games, then there are 8221 combinations of outcomes in which there is a team that wins all of its games. (Note that there cannot be two teams that win all of their games, since these two teams have to play a game.) Similarly, there are 8221 combinations of outcomes in which there is a team that loses all of its games. (Can you see why?)
    Before arriving at our conclusion, we note that there might be combinations of outcomes that are included in both of these counts. That is, there might be combinations of outcomes in which there is a team that wins all of its games and in which there is a team that loses all of its games.
    Since this total has been included in both sets of 8221 combinations of outcomes, we need to determine this total and subtract it once to leave these combinations included exactly once in our total.
    To determine the number of combinations of outcomes in this case, we choose a team (X) to win all of its games and a team (Y) to lose all of its games.
    Once X is chosen, the outcomes of its 7 games are all determined (X wins).
    Once Y is chosen, the outcomes of its 6 additional games are all determined (Y loses these 6 games plus the game with X that has already been determined).
    The outcomes of the remaining 2876=15 games are undetermined.
    Therefore, the number of combinations of outcomes is 87215 since there are 8 ways of choosing X, and then 7 ways of choosing Y (any team but X), and then 215 combinations of outcomes for the undetermined games.
    Thus, there are 8221+822187215 combinations of outcomes in which either one team loses 0 games or one team wins 0 games (or both).
    Therefore, the probability that one team loses 0 games or one team wins 0 games is 8221+822187215228=215(826+82687)228=2326+2326237213=26+267210 This means that the probability that every team loses at least one game and wins at least one game is 164+6471024=11211024=9031024.

    Answer: (D)

  25. Suppose that r=532+32.
    Thus, r2=532+32 and so 2r2=53+3 or 2r23=53.
    Squaring both sides again, we obtain (2r23)2=53 or 4r412r2+9=53 which gives 4r412r244=0 or r43r211=0 or r4=3r2+11.
    Suppose next that r100=2r98+14r96+11r94r50+ar46+br44+cr40() for some positive integers a,b,c.
    Since r0, we can divide by r40 to obtain r60=2r58+14r56+11r54r10+ar6+br4+c Now using the relationship r4=3r2+11, we can see that r602r5814r5611r54=r54(r62r414r211)=r54(r2(3r2+11)2r414r211)=r54(3r4+11r22r414r211)=r54(r43r211)=r54(0)=0 Therefore, the equation () is equivalent to the much simpler equation r10=ar6+br4+c Next, we express r10 and r6 as combinations of r2 and constant terms. (To do this, we will need to express r8 in this way too.) r6=r2r4=r2(3r2+11)=3r4+11r2=3(3r2+11)+11r2=20r2+33

    r8=r2r6=r2(20r2+33)=20r4+33r2=20(3r2+11)+33r2=93r2+220 r10=r2r8=r2(93r2+220)=93r4+220r2=93(3r2+11)+220r2=499r2+1023 Therefore, the equation r10=ar6+br4+c is equivalent to 499r2+1023=a(20r2+33)+b(3r2+11)+c Rearranging, we obtain 0=r2(20a+3b499)+(33a+11b+c1023) Therefore, if 20a+3b=499 and 33a+11b+c=1023, then the equation is satisfied. (It also turns out that if the equation is satisfied, then it must be the case that 20a+3b=499 and 33a+11b+c=1023. This is because r2 is an irrational number.)
    So the original problem is equivalent to finding positive integers a,b,c with 20a+3b=499 and 33a+11b+c=1023.
    We proceed by finding pairs (a,b) of positive integers that satisfy 20a+3b=499 and then checking to see if the value of c=102333a11b is positive. Since we need to find one triple (a,b,c) of positive integers, we do not have to worry greatly about justifying that we have all solutions at any step.
    Since 20a has a ones digit of 0 and 20a+3b=499, then the ones digit of 3b must be 9, which means that the ones digit of b must be 3.
    If b=3, we obtain 20a=4993b=490 and so a is not an integer.
    If b=13, we obtain 20a=4993b=460 and so a=23.
    Note that, from (a,b)=(23,13), we can obtain additional solutions by noticing that 20(3)=3(20) and so if we decrease a by 3 and increase b by 20, the sum 20a+3b does not change.
    However, it turns out that if (a,b)=(23,13), then c=102333(23)11(13)=121.
    Since we are only looking for a unique triple (a,b,c), then (a,b,c)=(23,13,121).
    Finally, a2+b2+c2=232+132+1212=15339.

    Answer: (D)