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2015 Canadian Team Mathematics Contest
Solutions

May 2015

© 2015 University of Waterloo

Individual Problems

  1. The inequality 5+3n>300 is equivalent to 3n>295, which is equivalent to n>2953.
    Since 2953=9813, then this is equivalent to n>9813.
    Therefore, the smallest positive integer n for which 5+3n>300 is n=99.

    Answer: 99

  2. We label the diagram as shown.

    The ladder that reaches 12m up the left wall is labelled as line AF, the other ladder is labelled as line DF. The left wall meets the ground at B, and the right wall meets the ground at C.

    Since AF=DF=15 m, then by the Pythagorean Theorem, BF=152122=81=9 mFC=15292=144=12 m Therefore, BC=BF+FC=9+12=21 m, and so the walls are 21 m apart.

    Answer: 21

  3. Let x be the number of friends who like to both ski and snowboard.
    Then 11x of the friends like to ski but not do not like to snowboard, and 13x of the friends like to snowboard but do not like to ski.
    Since 3 of the 20 friends do not like to ski or snowboard, then 17 like to either ski or snowboard or both.
    Thus, (11x)+(13x)+x=17 and so x=7.
    Therefore, 7 of the friends like to both ski and snowboard.

    Answer: 7

  4. Since (x,y)=(2,5) is the solution of the system of equations, then (x,y)=(2,5) satisfies both equations.
    Since (x,y)=(2,5) satisfies ax+2y=16, then 2a+10=16 or 2a=6 and so a=3.
    Since (x,y)=(2,5) satisfies 3xy=c, then 65=c or c=1.
    Therefore, ac=3.

    Answer: 3

  5. We note that 45=325 and so 45k=325k.
    For 45k to be a perfect square, each prime factor has to occur an even number of times.
    Therefore, k must be divisible by 5.
    We try the smallest two-digit possibilities for k that are divisible by 5, namely k=10,15,20,.
    If k=10, then 45k=450, which is not a perfect square.
    If k=15, then 45k=675, which is not a perfect square.
    If k=20, then 45k=900=302, which is a perfect square.
    Therefore, the smallest two-digit positive integer k for which 45k is a perfect square is k=20.

    Answer: 20

  6. Suppose that the distance that Clara has to travel is d km and that the time from when she starts to the scheduled meeting time is T hours.
    When she travels at an average speed of 20 km/h, she arrives half an hour early.
    Thus, she travels for T12 hours and so d20=T12.
    When she travels at an average speed of 12 km/h, she arrives half an hour late.
    Thus, she travels for T+12 hours and so d12=T+12.
    Adding these two equations, we obtain d20+d12=2T or 3d60+5d60=2T.
    Thus, 8d60=2T or d15=T.
    This means that if Clara travels at an average speed of 15 km/h, then it will take her exactly T hours to travel the d km, and so she will arrive at the scheduled time.

    Answer: 15

  7. Since the sum of any four consecutive terms is 17, then 8+b+c+d=17 or b+c+d=9.
    Also, b+c+d+e=17 and so 9+e=17 or e=8.
    Similarly, e+f+g+2=17 and d+e+f+g=17 tell us that d=2.
    But c+d+e+f=17 and e=8 and d=2, which gives c+f=1782=7.

    Answer: 7

  8. Suppose that the prism has AB=x, AD=y, and AF=z.
    Since the sum of all of the edge lengths is 24, then 4x+4y+4z=24 or x+y+z=6.
    (The prism has 4 edges of each length.)
    Since the surface area is 11, then 2xy+2xz+2yz=11.
    (The prism has 2 faces that are x by y, 2 faces that are x by z, and two faces that are y by z.)
    By the Pythagorean Theorem, AH2=AC2+CH2.

    Again, by the Pythagorean Theorem, AC2=AB2+BC2, and so AH2=AB2+BC2+CH2.
    But AB=x, BC=AD=y, and CH=AF=z.
    Therefore, AH2=x2+y2+z2.
    Finally, (x+y+z)2=(x+(y+z))2=x2+2x(y+z)+(y+z)2=x2+2xy+2xz+y2+2yz+z2 and so (x+y+z)2=x2+y2+z2+2xy+2xz+2yz, which gives AH2=(x+y+z)2(2xy+2xz+2yz)=6211=25 Since AH>0, then AH=25=5.

    Answer: 5

  9. Since the parabola opens upwards and has only one x-intercept, then its equation is of the form y=a(xr)2 for some real numbers a,r>0.
    Since the point A(1,4) is on the parabola, then (x,y)=(1,4) satisfies the equation of the parabola, which gives 4=a(1r)2.
    We will obtain a second equation for a and r by determining the coordinates of B.
    Since ABCD is a square, its diagonals AC and BD cross at their common midpoint, M, and are perpendicular at M.
    Since A has coordinates (1,4) and C has coordinates (394,374), then the coordinates of M are (12(1+394),12(4+374))=(438,538).
    To get from A to M, we go 358 units right and 218 units up.
    Thus, to get from M to B, we go 218 units right and 358 units down.
    This means that the coordinates of B are (438+218,538358)=(8,94).
    To see why this “over and up" approach is correct, draw a vertical line from M to P on the x-axis, and draw horizontal lines from A and B to points S and T, respectively, on MP.

    Since AS and BT are perpendicular to MP, then ASM and MTB are right-angled.
    Note that AM=MB, since ABCD is a square and M is its centre.
    Since AM and MB are perpendicular, then AMS+TMB=90.
    But SAM+AMS=90, so SAM=TMB.
    This means that ASM and MTB have equal corresponding angles.
    Since AM=MB, these triangles are congruent, and so AS=MT=358 and TB=SM=218.
    Thus, the coordinates of B are indeed (8,94).
    Since B is also on the parabola, then 94=a(8r)2.
    Dividing this equation by the first equation 4=a(1r)2, we obtain 916=(8r1r)2.
    Since 1<r<8 (the vertex of the parabola is between A and B), then 8r1r<0 and so 8r1r=34.
    Thus, 324r=3r3 and so 7r=35 or r=5.
    Since 4=a(1r)2, then 4=a(15)2 and so a=416=14.
    Therefore, the equation of the parabola is y=14(x5)2.

    Answer: y=14(x5)2

  10. First, we note that 257=256+1=28+1.
    Next, we note that 22008+1 is divisible by 28+1.
    This is because since 128+216224+21992+22000=1+(28)+(28)2+(28)3++(28)249+(28)250 and the right side is a geometric sequence with first term 1, common ratio 28, and 251 terms, giving 128+216224+21992+22000=1(1(28)251)1(28)=22008+128+1 Since the left side is an integer, then 22008+1 is divisible by 28+1=257.
    Since 22008+1 is divisible by 257, then any multiple of 22008+1 is divisible by 257.
    In particular, 27(22008+1)=22015+128 is divisible by 257.
    Since multiples of 257 differ by 257, then the largest multiple of 257 less than 22015+128 is 22015129 which is smaller than 22015.
    The next multiples of 257 larger than 22015+128 are 22015+385, 22015+642, 22015+899, and 22015+1156.
    Those between 22015 and 22015+1000 are 22015+128, 22015+385, 22015+642, 22015+899, which give values of N of 128, 385, 642, and 899.
    The sum of these values is 128+385+642+899=2054.

    Answer: 2054

Team Problems

  1. Re-arranging the terms and grouping into pairs, 1+3+5+7+9+11+13+15+17+19=(1+19)+(3+17)+(5+15)+(7+13)+(9+11)=520=100

    Answer: 100

  2. When the edge lengths of 10 are increased by 10%, the new edge lengths are 11.
    Therefore, the volume of the resulting cube is 113=1331.

    Answer: 1331

  3. Suppose that the cylinder has radius r.
    Since the circumference of its circular faces is 10π, then 2πr=10π or r=5.
    Therefore, the volume of the cylinder is πr2h=π524=100π.

    Answer: 100π

  4. Every prime number other than 2 is odd.
    Since a, b and c are all prime and a+b+c=22 which is even, it cannot be the case that all of a, b and c are odd (otherwise a+b+c would be odd).
    Thus, at least one of a, b and c is even.
    Since 1<a<b<c, then it must be the case that a=2 and b and c are odd primes.
    Since a=2, then b+c=20.
    Since b and c are primes with b<c, then b=3 and c=17, or b=7 and c=13.
    Therefore, there are two triples (a,b,c) of primes numbers that satisfy the requirements.

    Answer: 2

  5. Since TBD=110, then DBC=180TBD=180110=70.
    Since BCN=126, then DCB=180BCN=180126=54.
    Since the sum of the angles in DBC is 180, then BDC=1807054=56.
    Now BDC is an exterior angle of ADC.
    Thus, BDC=DAC+DCA.
    But ADC is isosceles with DAC=DCA.
    Thus, BDC=2DAC.
    Since BDC=56, then DAC=12(56)=28.
    Therefore, PAC=180DAC=18028=152.

    Answer: 152

  6. Note that 234=926=23213.
    Thus, k234=k23213.
    This product is divisible by 12=223 if and only if k contributes another factor of 2 (that is, if and only if k is even).
    Since k is a one-digit positive integer, then k=2,4,6,8.
    Therefore, there are 4 one-digit positive integers k for which k234 is divisible by 12.

    Answer: 4

  7. Since A(5,8), B(9,30), and C(n,n) lie on the same straight line, then the slopes of AB and AC are equal.
    Thus, (30)(8)95=n(8)n5224=n+8n522n+110=4n+3278=26nn=3 Therefore, n=3.

    Answer: 3

  8. To find the largest possible three-digit positive integer with no repeated digits, we put the largest possible digit in each position, starting with the hundreds digit (9), then the tens digit (8), and finally the ones (units) digit (7).
    Similarly, to find the smallest possible three-digit positive integer with no repeated digits, we put the smallest possible digit in each position, starting with the hundreds digit (1 because the leading digit cannot be 0), then the tens digit (0), and finally the ones (units) digit (2).
    The difference between the numbers 987 and 102 is 885.

    Answer: 885

  9. Since 0<x<y, then 2x+3y>2x+3x=5x.
    Since 2x+3y=80 and 2x+3y>5x, then 80>5x or 16>x.
    Since 2x+3y=80, then 2x=803y. Since 2x is even, then 803y is even. Since 80 is even, then 3y is even, and so y must be even.
    Set y=2Y for some positive integer Y.
    Then 2x+3y=80 becomes 2x+6Y=80 or x+3Y=40 or x=403Y.
    Since x<16, then 3Y>24 or Y>8.
    Since x>0, then 3Y<40 or Y<1313.
    When Y=9 (that is, y=18), x=13.
    When Y=10 (that is, y=20), x=10.
    When Y=11 (that is, y=22), x=7.
    When Y=12 (that is, y=24), x=4.
    When Y=13 (that is, y=26), x=1.
    Therefore, there are 5 pairs that satisfy the given conditions: (x,y)=(13,18),(10,20),(7,22), (4,24),(1,26).

    Answer: 5

  10. Since BAC=30 and BCA=90, then ABC is a 30-60-90 triangle.
    Thus, AB=2BC.
    But BC+AB=10 m (the height of the pole).
    Therefore, 3BC=10 m or BC=103 m.

    Answer: 103 m

  11. Since a=23 and b=32, then ab=89=1.
    Therefore, (ab)2015+12015(ab)201512015=(1)2015+1(1)20151=1+111=0

    Answer: 0

  12. For a positive integer to be a perfect square, its prime factors must occur in pairs.
    Since only one number from the list 1, 2, 3, 4, 5, 6, 7, 8, 9 is divisible by 5 (namely 5) and only one number from the list is divisible by 7, then neither 5 nor 7 can be one of the three integers chosen.
    Therefore, we need to choose three numbers from the list 1, 2, 3, 4, 6, 8, 9.
    We note that 982=144=122, which is a perfect square.
    Are there perfect squares larger than 144 that can be made?
    If the product of three integers is odd, then all three integers are odd. The only three integers in the list that are odd are 1, 3, 9. Their product is 27, which is not a perfect square.
    Therefore, any perfect square larger than 144 that can be made must be even.
    Further, note that the largest possible product of three numbers in the second list is 9×8×6 or 432, and so any perfect square that can be made is less than 432.
    The even perfect squares between 144 and 432 are 142=196, 162=256, 182=324, and 202=400.
    We cannot make 196 or 400, since no number in the list is a multiple of 7 or 5.
    We cannot obtain a product of 182=324=3422, since 324 has 4 factors of 3, which would require using 3, 6 and 9, whose product is too small.
    We cannot obtain a product of 162=256=28, since if we multiply all 7 numbers from the list, there are only 7 factors of 2 in total (from 2, 4, 6, 8).
    Therefore, 144 is the largest perfect square which is the product of three different one-digit positive integers.

    Answer: 144

  13. Suppose that the distance from Point A to Point B is d m, that Mario’s speed is w m/s, and that the speed of the moving sidewalk is v m/s.
    From the given information, dw=45 and dv=90.
    The amount of the time, in seconds, that it takes Mario to walk along the sidewalk when it is moving is dv+w, since his resulting speed is the sum of his walking speed and the speed of the sidewalk.
    Here, dv+w=1v+wd=1vd+wd=1190+145=1390=30 Therefore, it takes Mario 30 seconds to walk from Point A to Point B with the sidewalk moving.

    Answer: 30 seconds

  14. If a square has side length s, then the length of its diagonal is 2s.
    Since we are told that the length of the diagonal is s+1, then 2s=s+1 or 2ss=1 or (21)s=1.
    Thus, s=121=2+1(21)(2+1)=2+121=2+1.
    The area of the square is s2=(2+1)2=2+22+1=3+22.

    Answer: 3+22

  15. In the diagram, every path that starts at the top and follows to the bottom spells the word “WATERLOO".
    Therefore, we count the total number of paths from the top to the bottom.
    There is one path leading from the W to each of the A’s.
    The number of paths leading to a given letter below the A’s is the sum of the numbers of paths leading to the two letters above that point to the given letter.
    This is because each path leading to a given letter must go through exactly one of the letters above pointing to the given letter.
    We calculate the number of paths leading to each letter in the grid by proceeding downwards through the grid:

    A complete description of the diagram follows.

    Therefore, there are 20 paths that spell “WATERLOO".

    Answer: 20

  16. Using exponent and trigonometric laws, 4sin2x2cos2x=284(22)sin2x2cos2x=223422sin2x2cos2x=2123/422sin2x+cos2x=27/42sin2x+cos2x=74sin2x+(sin2x+cos2x)=74sin2x+1=74sin2x=34 Thus, sinx=±32.
    The smallest positive angle for which one of these is true is x=60 (or x=13π in radians).

    Answer: 60

  17. Using exponent laws, log3n6753=logn75(3n)log3n6753=(3n)logn756753=3logn75nlogn756753=3logn757593=3logn753231/2=3logn7535/2=3logn755/2=logn75n5/2=75n5=752 Therefore, n5=752=5625.

    Answer: 5625

  18. For the quadratic equation x2dx+e=0, the sum of the roots is d and the product of the roots is e.
    This is because if the roots are x=f and x=g, then x2dx+e=(xf)(xg)=x2fxgx+fg=x2(f+g)x+fg Suppose that r and s are the roots of x25x+2=0.
    Then r+s=5 and rs=2.
    We are told that the roots of x2+bx+c=0 are r2 and s2.
    Then c=r2s2=(rs)2=22=4 and b=r2+s2=(r+s)22rs=522(2)=21 which tells us that b=21.
    Therefore, cb=421=421.
    (We could also solve the problem by directly calculating the roots of x25x+2=0, squaring these roots, and creating a quadratic equation with these results as roots.)

    Answer: -421

  19. First, we note that (123)=12!3!9!=121110321=21110=220.
    This means that there are 220 different combinations of 3 eggs that can be chosen from 12 eggs.
    Zach randomly gives 3 eggs to each of the 4 children.
    The probability that a specific one of these 4 children receives the specific set of all 3 special eggs is 1220, since there are 220 different combinations of 3 eggs that can be given and each set is equally likely to be chosen.
    The probability that any one of these 4 children receives a specific set of 3 eggs is 41220=155 since the probability that each one receives the 3 special eggs is 1220 and these four events are disjoint.
    Therefore, the probability that only one child receives an egg that contains a prize is 155.

    Answer: 155

  20. If u0, then f(u)+f(1u)=u21+u2+1u21+1u2=u21+u2+1u2u2u2+1u2=u21+u2+1u2+1=u2+1u2+1=1 Therefore, A+B=(f(1)+f(1))+(f(2)+f(12))+(f(3)+f(13))++(f(2015)+f(12015)) and the right side equals 20151.
    Therefore, A+B=2015.

    Answer: 2015

  21. We first note that the coordinates of Pn+1 are (((n+1)1)2,(n+1)((n+1)1))=(n2,n(n+1)) and that the coordinates of Qn+1 are (((n+1)1)2,0)=(n2,0).
    Since Pn and Qn have the same x-coordinate, then PnQn is vertical.
    Since Pn+1 and Qn+1 have the same x-coordinate, then Pn+1Qn+1 is vertical.
    Since Qn and Qn+1 have the same y-coordinate, then QnQn+1 is horizontal.

    Let An be the area of trapezoid QnPnPn+1Qn+1.
    Then An=12(PnQn+Pn+1Qn+1)QnQn+1=12((n(n1)0)+(n(n+1)0))(n2(n1)2)=12(2n2)(2n1)=n2(2n1) We want to determine the number of integers n with 2n99 for which An is a perfect square.
    Since n2 is a perfect square, then An is a perfect square if and only if 2n1 is a perfect square.
    We note that 2n1 is odd and, since 2n99, then 32n1197.
    Thus, the possible perfect square values of 2n1 are the odd perfect squares between 3 and 197, which are 32,52,72,92,112,132 (that is, 9,25,49,81,121,169).
    Therefore, there are 6 integers n that satisfy the given conditions.

    Answer: 6

  22. Solution 1
    Let SR=x and the radius of the circle be r.
    Extend RP to meet the circle again at U.
    Then PS=PQ=PU=r, since each is a radius.

    By the Pythagorean Theorem in PQR, we have PQ2+PR2=QR2 or r2+(r+x)2=182.
    By the Secant-Secant Theorem, RSRU=RTRQ or x(x+2r)=10(18).
    From the first equation, r2+r2+2xr+x2=324.
    From the second equation, x2+2xr=180.
    Substituting, we obtain 2r2+180=324 or r2=72 and so r=72=62 since r>0.
    Since PQR is right-angled, then θ=PRQ is acute, which means that cosθ>0.
    Also, cos2θ=(PRRQ)2=RQ2PQ2RQ2=32472324=252324=79 Therefore, cosθ=79=73.

    Solution 2
    Let SR=x and the radius of the circle be r.
    Join PT. Then PQ=PT=r.
    Since QT=8 and TR=10, then QR=18.
    By the Pythagorean Theorem, PR=QR2PQ2=182r2.
    Therefore, cosθ=PRQR=182r218.
    By the cosine law in PTR, PT2=PR2+TR22(PR)(TR)cosθr2=(182r2)+1022182r2(10)182r218r2=324r2+100109(324r2)r2=424r2360+109r289r2=64r2=72 Therefore, cosθ=324r218=25218=73.

    Answer: 73

  23. Suppose that the largest integer in S is L, the smallest integer in S is P, and the mean of the elements of S is m.
    We are told that m=25L and m=74P.
    Thus, 74P=25L or P=835L.
    Since P and L are positive integers, then L must be divisible by 35.
    That is, L=35k for some positive integer k.
    From this, P=8k and m=14k.
    Now, the sum of the elements of S is nm=14kn.
    But all of the elements of S are positive and they include at least P=8k and L=35k, whose sum is 43k.
    Therefore, 14kn43k, which means that n4.
    If n=4, then the sum of the elements in S is 56k.
    This means that there are two elements in addition to 8k and 35k and the sum of these two additional elements is 56k43k=13k.
    But each of these elements must be at least 8k and so their sum is at least 16k.
    This is a contradiction, so there cannot be 4 elements in S.
    If n=5, then the sum of the elements in S is 70k.
    This means that there are three elements in addition to 8k and 35k and the sum of these three additional elements is 70k43k=27k.
    If n=5 and k=2, we have P=16 and L=70 and we want the sum of the remaining three elements, which must be distinct integers larger than 16 and less than 70, to be 54.
    This is possible with elements 17, 18, 19.
    In this case, S={16,17,18,19,70}. The sum of the elements of S is 140 and the average is 28. Note that 28=7416=2570.
    Therefore, the minimum possible value of n is 5.

    Answer: 5

  24. We cut the lateral surface of the cone along line segment IA.
    This forms a sector of a circle with radius IA=4.
    We label the second endpoint of the arc of this sector as A (it previously connected with A).
    Since the circumference of the base of the original cone is 2π(1)=2π, then the length of the arc of this sector is 2π.
    Since a full circle with radius 4 has circumference 8π and length of the arc of this sector is 2π, then this sector is 14 of a circle, so has central angle AIA=90.
    Point R still lies on IA with IR=3.
    The shortest path starting at R, travelling once around the cone, and ending at A is now the shortest path in this sector beginning at R and ending at A. This is the straight line segment from R to A.
    Since IR=3, IA=4, and RIA=90, then RA=32+42=25=5, by the Pythagorean Theorem.
    The point P will be the point on RA for which IP is perpendicular to RA.

    The unrolled cone looks like a quarter circle, with the bottom edge as IA', left edge as IA, and the outer edge (circumference of the circle) facing upwards to the right.

    We can then calculate the area of RIA in two different ways Area=12IRIA=12RAIP to obtain IRIA=RAIP or 34=5IP and so IP=125.
    Therefore, the length of IP is 125.
    (We could also calculate the length IP from RIA using trigonometry or similar triangles.)

    Answer: 125

  25. We label the regions with unknown integers u,v,w,x,y, as shown:

    The top left region is labelled as u, top right is v, bottom left is x, bottom right is y, and the centre region is w.

    The given list of integers includes 4 odd integers (1,3,5,9) and 7 even integers (2,4,6,8,10,12,14).
    We proceed by focusing on w and considering a number of cases:

    Therefore, in total there are 32+96+176+216=520 possible labellings.

    Answer: 520

Relay Problems

(Note: Where possible, the solutions to parts (b) and (c) of each Relay are written as if the value of t is not initially known, and then t is substituted at the end.)

    1. Evaluating, 2+0+1+5=8.

    2. The average of the numbers in the first list is m=12+15+9+14+105=605=12.
      The average of the numbers in the second list is n=24+t+8+124=44+t4=11+14t.
      Therefore, nm=(11+14t)12=14t1.
      Since the answer to (a) is 8, then t=8, and so nm=21=1.

    3. Since the two lines intersect at (a,b), then these coordinates satisfy the equation of each line.
      Therefore, b=13 and b=3a+t.
      Since b=13, then 13=3a+t or 3a=13t, and so a=13313t.
      Since the answer to (b) is 1, then t=1, and so a=13313=4.

    Answer: 8,1,4

    1. Since 1024=210 and 2k+4=1024, then k+4=10 or k=6.

    2. If 2t+2xt3x+4x+2t=30, then 3t+3x=30 or t+x=10 and so x=10t.
      Since the answer to (a) is 6, then t=6 and so x=10t=4.

    3. We use the Pythagorean Theorem repeatedly and obtain DE2=CD2+CE2=CD2+(BC2+BE2)=CD2+BC2+(AB2+AE2) Since AE=5, AB=4, BC=3, and CD=t, then DE2=t+3+4+5=t+12 Since the answer to (b) is 4, then t=4, and so DE2=16.
      Since DE>0, then DE=16=4.

    Answer: 6,4,4

    1. Since A and B have the same y-coordinate, then AB is horizontal.
      We treat AB as the base of ABC. It has length 5(1)=6.
      The height of ABC is the vertical distance from C(4,3) to the line through AB (which is y=2). This distance is 2(3)=5.
      Therefore, the area of ABC is 12(6)(5)=15.

    2. According to the given information, at last night’s rehearsal, Canada’s Totally Musical Choir (CTMC) spent 75630t=39t minutes singing their pieces.
      As a percentage of the full 75 minute rehearsal, this is 39t75×100%=(43(39t))%=(5243t)% Since the answer to (a) is 15, then t=15 and so CTMC spent (5243t)%=32% of their rehearsal singing their pieces, and so N=32.

    3. Using the rule that the sum of two entries in consecutive boxes in one row is the entry between these boxes in the row above, the entries in the second row from the bottom should be x+8,x+y,y+9,23.
      Thus, the first entry in the third row from the bottom should be 2x+y+8 and the last entry in this row should be y+32.
      Also, the two entries in the second row from the top should be 5x+4y+8 and 3x+4y+32.

      The given expressions are to be placed from left to right.

      Now the given entry 3(x+y) is the sum of the two entries below it.
      Thus, 3x+3y=(x+y)+(y+9), which gives 2x+y=9.
      Also, the top entry t is the sum of the two entries below it.
      Thus, t=(5x+4y+8)+(3x+4y+32), which gives 8x+8y=t40 or x+y=18t5.
      Since 2x+y=9 and x+y=18t5, then x=(2x+y)(x+y)=9(18t5)=1418t.
      Since x+y=18t5 and x=1418t, then y=(x+y)x=14t19.
      Thus, xy=(1418t)(14t19)=3338t.
      Since the answer to (b) is 32, then t=32 and so xy=3338t=21.

    Answer: 15,32,21

    1. A rectangular prism has six faces.
      Since the prism has edge lengths 2, 3 and 4, then the prism has two faces with dimensions 2×3 (each has area 6), two faces with dimensions 2×4 (each has area 8), and two faces with dimensions 3×4 (each has area 12).
      Therefore, the surface area of the prism is 2(6)+2(8)+2(12)=52.

    2. Since AB and CD are parallel, then FYZ=AVY=72 (alternating angles).
      Also, FZY=DZG=x (opposite angles).
      In FYZ, the sum of the angles is 180, and so YFZ+FYZ+FZY=180.
      Thus, t+72+x=180 or x=108t.
      Since the answer to (a) is 52, then t=52 and so x=108t=56.

    3. We do some preliminary work assuming that the number t to be received is a positive integer. (This turns out to be the case; if it were not the case, we might have to change strategies.)
      Since 1!=1, then 30t is divisible by 1! regardless of the value of t.
      Since 2!=2, then 30t=2(15t) is divisible by 2! regardless of the value of t.
      Since 3!=6, then 30t=6(15t) is divisible by 3! regardless of the value of t.
      Next, consider 4!=24.
      Since 30t4!=30t24=5t4, then 30t is divisible by 4! if and only if t is divisible by 4.
      Next consider 5!=120.
      Since 30t5!=30t120=t4, then 30t is divisible by 5! if and only if t is divisible by 4.
      Next consider 6!=720.
      Since 30t6!=30t720=t24, then 30t is divisible by 6! if and only if t is divisible by 24.
      Since the answer to (b) is 56, then t=56.
      Since t is divisible by 4 but not by 24, then 30t is divisible by 1!, 2!, 3!, 4!, and 5!, but not by 6!.
      Furthermore, 30t=1680 and if n7, then n!7!=5040, so 30t is not divisible by n! for n7 (because n!>30t).
      Therefore, there are 5 integers b for which 30t is divisible by b!.

    Answer: 52,56,5