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2015 Fermat Contest
Solutions
(Grade 11)

Tuesday, February 24, 2015
(in North America and South America)

Wednesday, February 25, 2015
(outside of North American and South America)

©2014 University of Waterloo


  1. The average of the five given numbers is 8+9+10+11+125=505=10.
    Alternatively, since there are an odd number of consecutive integers, the average is the middle number, which is 10.

    Answer: (E)

  2. Evaluating, 2×3+42+3=6+45=105=2.

    Answer: (A)

  3. Suppose that the distance between consecutive points on the path is d.
    Then, walking from P to U, Emily walks a total distance of 5d.
    Walking back from U to P, she also walks a distance of 5d, for a total distance walked of 10d.
    Since 70% of 10 is 7, then she has completed 70% of her walk after walking a distance of 7d (that is, after having walked 7 segments).
    She has walked a distance of 7d after walking a distance of 2d back from U, which gets her to S.

    Answer: (D)

  4. Evaluating, (x3)2=(33)2=(6)2=36.

    Answer: (B)

  5. Based on the diagram shown, the order of the vertices of the rectangle must be PQRS.

    A cartesian plane with the points Q, R, and P. They have the coordinates (3,1), (7,1), and (3,-2) respectively.

    (In any other configuration, either QRP or QPR would be an angle in the rectangle, which is not possible.)
    Since P and Q have the same x-coordinate, then side PQ of the rectangle is vertical.
    This means that side SR must also be vertical, and so the x-coordinate of S is the same as the x-coordinate of R, which is 7.

    Since Q and R have the same y-coordinate, then side QR of the rectangle is horizontal.
    This means that side PS must also be horizontal, and so the y-coordinate of S is the same as the y-coordinate of P, which is 2.
    Thus, the coordinates of S are (7,2).

    Answer: (B)

  6. Since MNPQ is a rectangle, then NMQ is a right angle and thus so is NMZ.
    The sum of the angles in NMZ is 180, so NZM=180ZNMZMN=1806890=22

    Since the sum of the angles in ZXY is also 180, then YXZ=180XZYXYZ=1802255=103

    Answer: (E)

  7. Violet starts with one-half of the money that she needed to buy the necklace.
    After her sister gives her money, she has three-quarters of the amount that she needs.
    This means that her sister gave her 3412=14 of the total amount that she needs.
    Since she now has three-quarters of the amount that she needs, then she still needs one-quarter of the total cost.
    In other words, her father will give her the same amount that her sister gave her, or $30.

    Answer: (D)

  8. Since 152=225 and 15=35, then 225=152=(35)2=3252.
    Therefore, x=2 and y=2, so x+y=4.

    Answer: (B)

  9. Solution 1
    The two teams include a total of 25+19=44 players.
    There are exactly 36 students who are at least one team.
    Thus, there are 4436=8 students who are counted twice.
    Therefore, there are 8 students who play both baseball and hockey.

    Solution 2
    Suppose that there are x students who play both baseball and hockey.
    Since there are 25 students who play baseball, then 25x of these play baseball and not hockey.
    Since there are 19 students who play hockey, then 19x of these play hockey and not baseball.
    Since 36 students play either baseball or hockey or both, then (25x)+(19x)+x=36 (The left side is the sum of the numbers of those who play baseball and not hockey, those who play hockey and not baseball, and those who play both.)
    Therefore, 44x=36 and so x=4436=8.
    Thus, 8 students play both baseball and hockey.

    Answer: (B)

  10. Since Bruce drove 200 km at a speed of 50 km/h, this took him 20050=4 hours.
    Anca drove the same 200 km at a speed of 60 km/h with a stop somewhere along the way.
    Since Anca drove 200 km at a speed of 60 km/h, the time that the driving portion of her trip took was 20060=313 hours.
    The length of Anca’s stop is the difference in driving times, or 4313=23 hours.
    Since 23 hours equals 40 minutes, then Anca stops for 40 minutes.

    Answer: (A)

  11. For each of the three digits of such a positive integer, there are three choices of digit (7, 8 or 9).
    Therefore, there are 333=27 possible integers that use no digits other than 7, 8 or 9.
    (We note that there are 9 such integers beginning with each of 7, 8 and 9.
    The 9 such integers beginning with 7 are 777, 778, 779, 787, 788, 789, 797, 798, 799.)

    Answer: (E)

  12. Since cos60=12 and cos45=12, then the given equation cos60=cos45cosθ becomes 12=12cosθ.
    Therefore, cosθ=22=12.
    Since 0θ90, then θ=45.

    Answer: (D)

  13. We make a table of the total amount of money that each of Steve and Wayne have at the end of each year. After the year 2000, each entry in Steve’s column is found by doubling the previous entry and each entry in Wayne’s column is found by dividing the previous entry by 2. We stop when the entry in Steve’s column is larger than that in Wayne’s column:
    Year Steve Wayne
    2000 $100 $10000
    2001 $200 $5000
    2002 $400 $2500
    2003 $800 $1250
    2004 $1600 $625

    Therefore, 2004 is the first time at which Steve has more money than Wayne at the end of the year.

    Answer: (C)

  14. Solution 1
    Since PQRS is a square, then its diagonal SQ cuts it into two equal areas.
    Therefore, the ratio of the area of PQS to the area of square PQRS is 1:2.
    PQS can be viewed as having base PS and height PQ.
    MQS can be viewed as having base MS and height PQ. (This is because PQ is perpendicular to the line containing MS.)
    Since MS=12PS, then the area of MQS is one-half of the area of PQS.
    Since the ratio of the area of PQS to the area of square PQRS is 1:2, then the ratio of the area of QMS to the area of square PQRS is 1:4.

    Solution 2
    Suppose that the side length of square PQRS is 2a.
    Then the area of square PQRS is (2a)2=4a2.
    Since M is the midpoint of side PS, then PM=MS=a.
    Then QMS can be seen as having base MS and height PQ. (This is because PQ is perpendicular to the line containing MS.)
    Since MS=a and PQ=2a, then the area of QMS is 12(MS)(PQ)=12a(2a)=a2.
    Therefore, the ratio of the area of QMS to the area of square PQRS is a2:4a2 which equals 1:4.

    Answer: (B)

  15. Solution 1
    Zoltan answered 45 questions.
    If all of his answers had been correct, his score would have been 45(4)=180 points.
    Since his score was 135 points, then he lost 180135=45 points.
    For each incorrect answer, he lost 5 points compared to a correct answer, since a correct answer adds 4 points and an incorrect answer subtracts 1 point.
    Therefore, Zoltan had 45÷5=9 incorrect answers.
    (We can check that 36 correct, 9 incorrect, and 5 unanswered gives a score of 4(36)1(9)+0(5) or 135 points.)

    Solution 2
    Suppose that Zoltan answered x questions incorrectly.
    Since he answered 45 questions in total, then he answered 45x questions correctly.
    Since the test included 50 questions and he answered 45, then he did not answer 5 questions.
    Using the marking scheme, his score was 4(45x)1(x)+0(5).
    We are told that his score was 135 points.
    Hence, 4(45x)1(x)+0(5)=135 and so 1804xx=135.
    Thus, 5x=45 or x=9.
    Therefore, Zoltan answered 9 questions incorrectly.
    (We can check that 36 correct, 9 incorrect, and 5 unanswered gives a score of 4(36)1(9)+0(5) or 135 points.)

    Answer: (A)

  16. Since P(4,0) and Q(16,0) are endpoints of a diameter of the semi-circle, then the length of the diameter is 16(4)=20.
    Since a diameter of the semi-circle has length 20, then the radius of the semi-circle is 12(20)=10.
    Also, the centre C is the midpoint of diameter PQ and so has coordinates (12(4+16),12(0+0)) or (6,0).

    Now the distance between C(6,0) and R(0,t) is 10, since CR is a radius.
    Therefore, (60)2+(0t)2=1036+t2=100t2=64
    (Alternatively, we could have noted that if O is the origin, then ROC is right-angled with RO=t, RC=10 and OC=6 and then used the Pythagorean Theorem to obtain t2+62=102, which gives t2=64.)
    Since t>0, then t=8.

    Answer: (C)

  17. Since a+bab=3, then a+b=3(ab) or a+b=3a3b.
    Thus, 4b=2a and so 2b=a or 2=ab.
    (Note that b0, since otherwise the original equation would become aa=3, which is not true.)

    Answer: (D)

  18. The equation x2+2kx+7k10=0 has two equal real roots precisely when the discriminant of this quadratic equation equals 0.
    The discriminant, Δ, equals Δ=(2k)24(1)(7k10)=4k228k+40 For the discriminant to equal 0, we have 4k228k+40=0 or k27k+10=0 or (k2)(k5)=0.
    Thus, k=2 or k=5.
    We check that each of these values gives an equation with the desired property.
    When k=2, the equation is x2+4x+4=0 which is equivalent to (x+2)2=0 and so only has one solution for x.
    When k=5, the equation is x2+10x+25=0 which is equivalent to (x+5)2=0 and so only has one solution for x.
    The sum of these values of k is 2+5=7.

    Answer: (E)

  19. Suppose the slope of the three parallel lines is m.
    The equation of a line with slope m and y-intercept 2 is y=mx+2.
    To find the x-intercept in terms of m, we set y=0 and solve for x.
    Doing this, we obtain mx+2=0 or x=2m.
    Similarly, the line with slope m and y-intercept 3 has x-intercept 3m.
    Also, the line with slope m and y-intercept 4 has x-intercept 4m.
    Since the sum of the x-intercepts of these lines is 36, then (2m)+(3m)+(4m)=36.
    Multiplying both sides by m, we obtain 234=36m and so 36m=9 or m=14.

    Answer: (E)

  20. First, we factor a2014+a2015 as a2014(1+a).
    If a=5 or a=10, then the factor a2014 is a multiple of 5, so the original expression is divisible by 5.
    If a=4 or a=9, then the factor (1+a) is a multiple of 5, so the original expression is divisible by 5.
    If a=1,2,3,6,7,8, then neither a2014 nor (1+a) is a multiple of 5.
    Since neither factor is a multiple of 5, which is a prime number, then the product a2014(1+a) is not divisible by 5.
    Therefore, there are four integers a in the range 1a10 for which a2014+a2015 is divisible by 5.

    Answer: (C)

  21. If Amina wins, she can win on her first turn, on her second turn, or on her third turn.
    If she wins on her first turn, then she went first and tossed tails.
    This occurs with probability 12.
    If she wins on her second turn, then she tossed heads, then Bert tossed heads, then Amina tossed tails. This gives the sequence HHT. The probability of this sequence of tosses occurring is 121212=18. (Note that there is only one possible sequence of Ts and Hs for which Amina wins on her second turn, and the probability of a specific toss on any turn is 12.)
    Similarly, if Amina wins on her third turn, then the sequence of tosses that must have occurred is HHHHT, which has probability (12)5=132.
    Therefore, the probability that Amina wins is 12+18+132=16+4+132=2132.

    Answer: (A)

  22. Since a, b and c form an arithmetic sequence in this order, then a=bd and c=b+d for some real number d.
    We note that d0, since otherwise we would have a=b=c and then abc=17955 would tell us that b3=17955 or b=179553, which is not an integer.
    Writing the terms of the geometric sequence in terms of b and d, we have 3a+b=3(bd)+b=4b3d3b+c=3b+(b+d)=4b+d3c+a=3(b+d)+(bd)=4b+2d Since 3a+b, 3b+c and 3c+a form a geometric sequence in this order, then 3b+c3a+b=3c+a3b+c(3b+c)2=(3a+b)(3c+a)(4b+d)2=(4b3d)(4b+2d)16b2+8bd+d2=16b24bd6d212bd=7d212b=7d(since d0)d=127b Therefore, a=bd=b(127b)=197b and c=b+d=b+(127b)=57b.
    Since abc=17955, then (197b)(b)(57b)=17955 or 9549b3=17955 or b3=9261 and so b=21.
    Thus, a=197b=197(21)=57 and c=57b=57(21)=15.
    We can check that a=57, b=21 and c=15 have a product of 17955, that 57,21,15 is indeed an arithmetic sequence (with common difference 36), and that 3a+b=192,3b+c=48, and 3c+a=12 form a geometric sequence (with common ratio 14).
    Therefore, a+b+c=(57)+(21)+15=63.

    Answer: (A)

  23. Starting from the given equation, we obtain the equivalent equations 5x24xy+2x+y2=6245x24xy+2x+y2+1=6254x24xy+y2+x2+2x+1=625(2xy)2+(x+1)2=625 Note that 625=252.
    Since x and y are both integers, then the left side of the given equation is the sum of two perfect squares. Since any perfect square is non-negative, then each of these perfect squares is at most 625=252.
    The perfect squares from 02 to 252 are: 0,1,4,9,16,25,36,49,64,81,100,121,144, 169,196,225,256,289,324,361,400,441,484,529,576,625 The pairs of perfect squares from this list that have a sum of 625 are 625=625+0=576+49=400+225 (We can verify this relatively quickly by checking the difference between 625 and each of the perfect squares in the list from 324 to 625 to see if this difference is itself a perfect square. We do not need to check anything less than 324, since 625 is odd and so one of the two squares adding to 625 must be larger than the other and hence larger than half of 625.)
    Therefore, (2xy)2 and (x+1)2 equal 252 and 02 in some order, or 242 and 72 in some order, or 202 and 152 in some order.
    Thus, 2xy and x+1 equal ±25 and 0 in some order, or ±24 and ±7 in some order, or ±20 and ±15 in some order.
    Since x0, then x+11, so we need to consider the possibilities that x+1=25,24,7,20,15:

    From this list, the pairs of non-negative integers (x,y) that satisfy the condition 0xy are (x,y)=(24,48),(23,39),(23,53),(6,36),(19,23),(19,53),(14,48).
    There are 7 such pairs. (We can check by direct substitution that each pair satisfies the original equation.)

    Answer: (E)

  24. Let r be the radius of the lower circle.
    We label the square as ABCD, the centre of the upper circle as U, and the centre of the lower circle as L. We call E the point at which the upper circle touches the top line, G the point at which the lower circle touches the bottom line, F the point at which the two circles touch, and H the point at which the lower circle touches the square.
    We join EU, UF, FL, LG, LH, and UA.
    We need to use two facts about circles:

    Extend EU downwards to meet LH at J. Since EU and AD are perpendicular to the parallel lines and LH is perpendicular to AD, then EJ is perpendicular to LH.
    Extend BA to meet EJ at K. Using a similar argument, AK is perpendicular to EJ.
    We focus on UJL, which is right-angled at J.
    Since the radius of the upper circle is 65, then we know that EU=UA=UF=65.
    Since the radius of the lower circle is r, then we know that FL=LH=LG=r.
    Thus, UL=UF+FL=65+r=r+65.

    Since the top and bottom lines are parallel, EJ and LG are perpendicular to these lines, and LJ is parallel to these lines, then EU+UJ+LG equals the distance between the lines.
    Thus, 65+UJ+r=400 and so UJ=335r.
    Next, we note that LJ=LHJH=rJH=rAK, since AKJH is a rectangle.
    Now UKA is right-angled and has UA=65.
    Also, UK=EKEU=EK65.
    But EK equals the difference between the distance between the lines and the side length of the square, or 400279=121.
    Thus, UK=12165=56.
    Using the Pythagorean Theorem in UKA, we obtain AK2=UA2UK2=652562 and so AK2=1089=332.
    Since AK>0, then AK=33.
    Thus, LJ=r33.
    Finally, using the Pythagorean Theorem in UJL, we obtain UJ2+LJ2=UL2(335r)2+(r33)2=(r+65)2r2670r+3352+r266r+332=r2+130r+652r2866r+109089=0 By the quadratic formula, r=866±86624(1)(109089)2=866±5602=153 or 713.
    Since r must be less than the distance between the two lines, which is 400, then r=153. Of the given answers, this is closest to 153.

    Answer: (C)

  25. This solution is written in as mathematically complete a way as possible. Of course, while doing a multiple choice contest, those who attempt and/or complete this problem would be unlikely to work through all of these details.

    Step 1: Using information about decimal equivalents to rewrite fractions
    Consider a real number x whose decimal equivalent is of the form 0.g1g2gpr1r2rq for some integers p0 and q>0 and digits g1,g2,,gp,r1,r2,,rq.
    (Note that if p=0, then 0.g1g2gpr1r2rq=0.r1r2rq.)
    Then x=c10p(10q1) for some positive integer c.
    We demonstrate this in a specific example and leave the full algebraic derivation to the end of this solution. If x=0.12745, then x=0.12745100x=12.745102x12=0.7451000(102x12)=745.745103(102x12)745=0.745103(102x12)745=102x12103(102x12)(102x12)=745(1031)(102x12)=745102x12=7451031102x=12+7451031x=12102+745102(1031)x=(1031)12+745102(1031) Suppose that a fraction mn has m and n positive integers with m<n. Then 0<mn<1.
    Suppose now that a fraction mn of positive integers with 0<mn<1 has the property that there is a sequence of consecutive digits of length 6 in its decimal equivalent that repeats consecutively and indefinitely. That is, suppose that mn=0.g1g2gpr1r2r6 for some integer p0 and digits g1,g2,,gp,r1,r2,r3,r4,r5,r6.
    From above, mn=c10p999999 for some positive integer c. (Note that 1061=999999.)

    Step 2: Using further conditions given to analyze n
    Continuing from above, cn=10p999999m.
    Since we are also told that mn is in lowest terms, then m and n have no common divisors larger than 1, and so n must be a divisor of 10p999999.
    Note that 10p999999=2p5p9991001=2p5p(3337)(1191)=2p5p333711713.
    Since n is a divisor of 2p5p333711713, then n cannot contain any prime factors other than 2,3,5,7,11,13,37.
    Since n is not divisible by the square of any positive integer, then it cannot be divisible by the square of any prime number.
    Thus, n must be a divisor of 2533711713=1111110.

    Step 3: Consolidating current progress
    We now know that any fraction mn satisfying the properties

    can be written in the form mn=s1111110 for some positive integer s with 1s1111109.
    (We note that we have not yet determined whether the shortest sequence of consecutive digits that repeats consecutively and indefinitely has length 6.)

    Step 4: Every fraction s1111110 with 1s1111109 can be written as a fraction satisfying these four bullets
    Each s1111110 is between 0 and 1, can be written in lowest terms and has denominator not divisible by the square of any positive integer larger than 1, so any equivalent fraction in lower (or lowest terms) shares this property as factors of the denominator will only be removed in reducing, not added.
    Furthermore, s1111110=1109s999999=110(y+z999999) for some non-negative integers y and z with 0y<10 and 0z<999998. (y and z are the quotient and remainder, respectively, when 9s is divided by 999999.)
    Writing z=r1r2r3r4r5r6 for some digits r1,r2,r3,r4,r5,r6 (some or all possibly 0), then s1111110=110(y+r1r2r3r4r5r6999999)=110(y.r1r2r3r4r5r6)=0.yr1r2r3r4r5r6 so every s1111110 can be written as a decimal with a repeating sequence of length 6.
    Also, each s1111110 is different and so will produce a different mn.
    Therefore, the number of such fractions s1111110 (which is 1111109) will equal the number of fractions mn that satisfy the four bullets above.
    Notice that we have not yet checked to see if the sequence of digits of length 6 is the shortest such sequence.

    Step 5: Considering shorter possible lengths
    Since 6 is to be the length of the shortest sequence of repeating digits after the decimal point, then there can be no sequence of repeating digits of length 1, 2, 3, 4, or 5.
    Using a similar approach to the first derivation above, we see that mn thus cannot be written in any of the forms c10p9 or c10p99 or c10p999 or c10p9999 or c10p99999.
    Using a similar approach to the analysis of prime factors above, we see that mn cannot be written in any of the forms t253=t30 or t25311=t330 or t25337=t1110 or t25311101=t33330 or t25341271=t333330.
    It is possible that an mn with properties as above including a repeating sequence of length 6 in its decimal equivalent can also be written with a repeating sequence of length 1, 2 or 3.
    This is because, for example, 0.r1=0.r1r1r1r1r1r10.r1r2=0.r1r2r1r2r1r20.r1r2r3=0.r1r2r3r1r2r3 which all have sequences of length 6 that repeat.
    It is not possible for an mn with a repeating sequence of length 6 in its decimal equivalent to also be written with a repeating sequence of length 4 or 5, without a repeating sequence of length 1 or 2.
    This is because if, for example, there is a repeating sequence of length 4, then s1111110=s2533711713=t25311101 for some positive integer t and so 25311101s=2533711713t and so 101s=37713t which tells us that 101 is divisor of t and so s1111110=t25311=t330 for some positive integer t.
    Therefore, the decimal equivalent to s1111110=t330 has a sequence of length 2 that repeats, so any fraction with a decimal equivalent that has a repeating sequence of length 4 will be dealt with among those with sequences of length 1, 2 or 3.
    In a similar way, we can rule out decimal equivalents with sequences of length 5 that repeat.
    Therefore, we need to (carefully) remove fractions from our count that have decimal equivalents with sequences of lengths 1, 2 or 3 that repeat.

    Step 6: Considering overlap
    We have 1111109 fractions mn to consider, each of which satisfies the four bullets above.
    Since the decimal equivalent of mn cannot have sequences of lengths 1, 2 or 3 that repeat, then mn cannot be written in any of the forms u1110 or v330 or w30 for positive integers u, v and w with u<1110, v<330, and w<30.
    Let U be the set of the 1009 fractions of the form u1110, V be the set of the 329 fractions of the form v330, and W be the set of the 29 fractions of the form w30.
    Any fraction of the form w30 is also of the form u1110 (since w30=37w1110) and is also of the form v330 (since w30=11w330).
    Therefore, every fraction in W is also in U and in V. In set notation WU and WV.
    Furthermore, any fraction that is in both U and V is also in W:

    Suppose that a fraction can be written in both forms u1110 and v330.
    Then u1110=v330 or u37=v11 and so 11u=37v.
    Since 37v is a thus a multiple of 11 and 37 is not divisible by the prime number 11, then v is a multiple of 11.
    This means that v330=11f330=f30 for some positive integer f, and so is in W.

    In set notation, UVW.
    Since WU and WV and UVW, then UV=W; in other words, the set of fractions in each of U and V is precisely the set of fractions W.

    Step 7: Final counting
    We start with 1111109 fractions, as above, and want to remove all of the fractions in U, V and W.
    Since each fraction in W is in U and V, it is enough to remove those U and V only.
    The total number of fractions in U and V (that is, in UV) equals the number of fractions in U plus the number of fractions in V minus the number of fractions in their overlap (that is, in UV=W). This is because any fraction in the overlap is “counted twice" when include all fractions in U and all fractions in V.
    Therefore, we need to remove 1009+32929 fractions from the set of 1111109.
    Therefore, F, the number of fractions having the desired properties, is F=1111109(1009+32929)=1109700 Since F has 7 digits, then G=F+7=1109707. The sum of the squares of the digits of G is 12+12+02+92+72+02+72=1+1+81+49+49=181.

    Step 8: General algebraic derivation from Step 1
    Consider a real number x whose decimal equivalent is of the form 0.g1g2gpr1r2rq for some integers p0 and q>0 and digits g1,g2,,gp,r1,r2,,rq. Then x=0.g1g2gpr1r2rq10px=g1g2gp.r1r2rq10pxg1g2gp=0.r1r2rq10q(10pxg1g2gp)=r1r2rq.r1r2rq10q(10pxg1g2gp)r1r2rq=0.r1r2rq10q(10pxg1g2gp)r1r2rq=10pxg1g2gp10q(10pxg1g2gp)(10pxg1g2gp)=r1r2rq(10q1)(10pxg1g2gp)=r1r2rq10pxg1g2gp=r1r2rq10q110px=g1g2gp+r1r2rq10q1x=g1g2gp10p+r1r2rq10p(10q1)x=(10q1)g1g2gp+r1r2rq10p(10q1)

    Answer: (E)