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2020 Euclid Contest
Solutions

Tuesday, April 7, 2020
(in North America and South America)

Wednesday, April 8, 2020
(outside of North American and South America)

©2020 University of Waterloo


    1. Solution 1:

      If x2, then 3x+6x+2=3(x+2)x+2=3.
      In other words, for every x2, the expression is equal to 3.
      Therefore, when x=11, we get 3x+6x+2=3.

      Solution 2:

      When x=11, we obtain 3x+6x+2=3(11)+611+2=3913=3.

    2. Solution 1:

      The point at which a line crosses the y-axis has x-coordinate 0.
      Because A has x-coordinate 1 and B has x-coordinate 1, then the midpoint of AB is on the y-axis and is on the line through A and B, so is the point at which this line crosses the x-axis.
      The midpoint of A(1,5) and B(1,7) is (12(1+1),12(5+7)) or (0,6).
      Therefore, the line that passes through A(1,5) and B(1,7) has y-intercept 6.

      Solution 2:

      The line through A(1,5) and B(1,7) has slope 751(1)=22=1.
      Since the line passes through B(1,7), its equation can be written as y7=1(x1) or y=x+6.
      The line with equation y=x+6 has y-intercept 6.

    3. First, we find the coordinates of the point at which the lines with equations y=3x+7 and y=x+9 intersect.
      Equating values of y, we obtain 3x+7=x+9 and so 2x=2 or x=1.
      When x=1, we get y=x+9=10.
      Thus, these two lines intersect at (1,10).
      Since all three lines pass through the same point, the line with equation y=mx+17 passes through (1,10).
      Therefore, 10=m1+17 which gives m=1017=7.

    1. Suppose that m has hundreds digit a, tens digit b, and ones (units) digit c.
      From the given information, a, b and c are distinct, each of a, b and c is less than 10, a=bc, and c is odd (since m is odd).
      The integer m=623 satisfies all of these conditions. Since we are told there is only one such number, then 623 must be the only answer.
      Why is this the only possible value of m?
      We note that we cannot have b=1 or c=1, otherwise a=c or a=b.
      Thus, b2 and c2.
      Since c2 and c is odd, then c can equal 3, 5, 7, or 9.
      Since b2 and a=bc, then if c equals 5, 7 or 9, a would be larger than 10, which is not possible.
      Thus, c=3.
      Since b2 and bc, then b=2 or b4.
      If b4 and c=3, then a>10, which is not possible.
      Therefore, we must have c=3 and b=2, which gives a=6.

    2. Since Eleanor has 100 marbles which are black and gold in the ratio 1:4, then 15 of her marbles are black, which means that she has 15100=20 black marbles.
      When more gold marbles are added, the ratio of black to gold is 1:6, which means that she has 620=120 gold marbles.
      Eleanor now has 20+120=140 marbles, which means that she added 140100=40 gold marbles.

    3. First, we see that n2+n+15n=n2n+nn+15n=n+1+15n.
      This means that n2+n+15n is an integer exactly when n+1+15n is an integer.
      Since n+1 is an integer, then n2+n+15n is an integer exactly when 15n is an integer.
      The expression 15n is an integer exactly when n is a divisor of 15.
      Since n is a positive integer, then the possible values of n are 1, 3, 5, and 15.

    1. First, we note that a triangle with one right angle and one angle with measure 45 is isosceles.
      This is because the measure of the third angle equals 1809045=45 which means that the triangle has two equal angles.
      In particular, CDE is isosceles with CD=DE and EFG is isosceles with EF=FG.
      Since DE=EF=1 m, then CD=FG=1 m.
      Join C to G.

      Consider quadrilateral CDFG. Since the angles at D and F are right angles and since CD=GF, it must be the case that CDFG is a rectangle.
      This means that CG=DF=2 m and that the angles at C and G are right angles.
      Since CGF=90 and DCG=90, then BGC=1809045=45 and BCG=90.
      This means that BCG is also isosceles with BC=CG=2 m.
      Finally, BD=BC+CD=2 m+1 m=3 m.

    2. We apply the process two more times:

      x y
      Before Step 1 24 3
      After Step 1 27 3
      After Step 2 81 3
      After Step 3 81 4
      x y
      Before Step 1 81 4
      After Step 1 85 4
      After Step 2 340 4
      After Step 3 340 5

      Therefore, the final value of x is 340.

    3. The parabola with equation y=kx2+6x+k has two distinct x-intercepts exactly when the discriminant of the quadratic equation kx2+6x+k=0 is positive.
      Here, the disciminant equals Δ=624kk=364k2.
      The inequality 364k2>0 is equivalent to k2<9.
      Since k is an integer and k0, then k can equal 2,1,1,2.
      (If k3 or k3, we get k29 so no values of k in these ranges give the desired result.)

    1. Since ab<47 and 47<1, then ab<1.
      Since a and b are positive integers, then a<b.
      Since the difference between a and b is 15 and a<b, then b=a+15.
      Therefore, we have 59<aa+15<47.
      We multiply both sides of the left inequality by 9(a+15) (which is positive) to obtain 5(a+15)<9a from which we get 5a+75<9a and so 4a>75.
      From this, we see that a>754=18.75.
      Since a is an integer, then a19.
      We multiply both sides of the right inequality by 7(a+15) (which is positive) to obtain 7a<4(a+15) from which we get 7a<4a+60 and so 3a<60.
      From this, we see that a<20.
      Since a is an integer, then a19.
      Since a19 and a19, then a=19, which means that ab=1934.

    2. The first 6 terms of a geometric sequence with first term 10 and common ratio 12 are 10,5,52,54,58,516.
      Here, the ratio of its 6th term to its 4th term is 5/165/4 which equals 14. (We could have determined this without writing out the sequence, since moving from the 4th term to the 6th involves multiplying by 12 twice.)
      The first 6 terms of an arithmetic sequence with first term 10 and common difference d are 10,10+d,10+2d,10+3d,10+4d,10+5d.
      Here, the ratio of the 6th term to the 4th term is 10+5d10+3d.
      Since these ratios are equal, then 10+5d10+3d=14, which gives 4(10+5d)=10+3d and so 40+20d=10+3d or 17d=30 and so d=3017.

    1. Let a=f(20). Then f(f(20))=f(a).
      To calculate f(f(20)), we determine the value of a and then the value of f(a).
      By definition, a=f(20) is the number of prime numbers p that satisfy 20p30.
      The prime numbers between 20 and 30, inclusive, are 23 and 29, so a=f(20)=2.
      Thus, f(f(20))=f(a)=f(2).
      By definition, f(2) is the number of prime numbers p that satisfy 2p12.
      The prime numbers between 2 and 12, inclusive, are 2, 3, 5, 7, 11, of which there are 5.
      Therefore, f(f(20))=5.

    2. Since (x1)(y2)=0, then x=1 or y=2.
      Suppose that x=1. In this case, the remaining equations become: (13)(z+2)=01+yz=9 or 2(z+2)=0yz=8 From the first of these equations, z=2.
      From the second of these equations, y(2)=8 and so y=4.
      Therefore, if x=1, the only solution is (x,y,z)=(1,4,2).
      Suppose that y=2. In this case, the remaining equations become: (x3)(z+2)=0x+2z=9 From the first equation x=3 or z=2.
      If x=3, then 3+2z=9 and so z=3.
      If z=2, then x+2(2)=9 and so x=13.
      Therefore, if y=2, the solutions are (x,y,z)=(3,2,3) and (x,y,z)=(13,2,2).
      In summary, the solutions to the system of equations are (x,y,z)=(1,4,2),(3,2,3),(13,2,2) We can check by substitution that each of these triples does indeed satisfy each of the equations.

    1. Draw a perpendicular from S to V on BC.
      Since ASVB is a quadrilateral with three right angles, then it has four right angles and so is a rectangle.
      Therefore, BV=AS=r, since AS is a radius of the top semi-circle, and SV=AB=4.
      Join S and T to P. Since the two semi-circles are tangent at P, then SPT is a straight line, which means that ST=SP+PT=r+r=2r.

      Consider right-angled SVT. We have SV=4 and ST=2r.
      Also, VT=BCBVTC=6rr=62r.
      By the Pythagorean Theorem, SV2+VT2=ST242+(62r)2=(2r)216+3624r+4r2=4r252=24r Thus, r=5224=136.

    2. Since ABE is right-angled at A and is isosceles with AB=AE=72, then ABE is a 45-45-90 triangle, which means that ABE=45 and BE=2AB=272=14.
      Since BCD is right-angled at C with DBDC=8x4x=2, then BCD is a 30-60-90 triangle, which means that DBC=30.
      Since ABC=135, then EBD=ABCABEDBC=1354530=60.
      Now consider EBD. We have EB=14, BD=8x, DE=8x6, and EBD=60.
      Using the cosine law, we obtain the following equivalent equations: DE2=EB2+BD22EBBDcos(EBD)(8x6)2=142+(8x)22(14)(8x)cos(60)64x296x+36=196+64x22(14)(8x)1296x=16014(8x)112x96x=16016x=160x=10 Therefore, the only possible value of x is x=10.

    1. Solution 1:

      Since the function g is linear and has positive slope, then it is one-to-one and so invertible.
      This means that g1(g(a))=a for every real number a and g(g1(b))=b for every real number b.
      Therefore, g(f(g1(g(a))))=g(f(a)) for every real number a.
      This means that g(f(a))=g(f(g1(g(a))))=2(g(a))2+16g(a)+26=2(2a4)2+16(2a4)+26=2(4a216a+16)+32a64+26=8a26 Furthermore, if b=f(a), then g1(g(f(a)))=g1(g(b))=b=f(a).
      Therefore, f(a)=g1(g(f(a)))=g1(8a26) Since g(x)=2x4, then y=2g1(y)4 and so g1(y)=12y+2.
      Therefore, f(a)=12(8a26)+2=4a21 and so f(π)=4π21.

      Solution 2:

      Since the function g is linear and has positive slope, then it is one-to-one and so invertible.
      To find a formula for g1(y), we start with the equation g(x)=2x4, convert to y=2g1(y)4 and then solve for g1(y) to obtain 2g1(y)=y+4 and so g1(y)=y+42.
      We are given that g(f(g1(x)))=2x2+16x+26.
      We can apply the function g1 to both sides to obtain successively: f(g1(x))=g1(2x2+16x+26)f(g1(x))=(2x2+16x+26)+42(knowing a formula for g1)f(g1(x))=x2+8x+15f(x+42)=x2+8x+15(knowing a formula for g1)f(x+42)=x2+8x+161f(x+42)=(x+4)21 We want to determine the value of f(π).
      Thus, we can replace x+42 with π, which is equivalent to replacing x+4 with 2π.
      Thus, f(π)=(2π)21=4π21.

    2. Solution 1:

      Using logarithm laws, the given equations are equivalent to log2(sinx)+log2(cosy)=32log2(sinx)log2(cosy)=12 Adding these two equations, we obtain 2log2(sinx)=1 which gives log2(sinx)=12 and so sinx=21/2=121/2=12.
      Since 0x<180, then x=45 or x=135.
      Since log2(sinx)+log2(cosy)=32 and log2(sinx)=12, then log2(cosy)=1, which gives cosy=21=12.
      Since 0y<180, then y=60.
      Therefore, (x,y)=(45,60) or (x,y)=(135,60).

      Solution 2:

      First, we note that 21/2=2 and 23/2=123/2=12121/2=122.
      From the given equations, we obtain sinxcosy=23/2=122sinxcosy=21/2=2 Multiplying these two equations together, we obtain (sinx)2=12 which gives sinx=±12.
      Since 0x<180, it must be the case that sinx0 and so sinx=12.
      Since 0x<180, we obtain x=45 or x=135.
      Since sinxcosy=122 and sinx=12, we obtain cosy=12.
      Since 0y<180, then y=60.
      Therefore, (x,y)=(45,60) or (x,y)=(135,60).

    1. Solution 1:

      Let x be the probability that Bianca wins the tournament.
      Because Alain, Bianca and Chen are equally matched and because their roles in the tournament are identical, then the probability that each of them wins will be the same.
      Thus, the probability that Alain wins the tournament is x and the probability that Chen wins the tournament is x.
      Let y be the probability that Dave wins the tournament.
      Since exactly one of Alain, Bianca, Chen, and Dave wins the tournament, then 3x+y=1 and so x=1y3. We can calculate y in terms of p.
      In order for Dave to win the tournament, he needs to win two matches.
      No matter who Dave plays, his probability of winning each match is p.
      Thus, the probability that he wins his two consecutive matches is p2 and so the probability that he wins the tournament is y=p2.
      Thus, the probability that Bianca wins the tournament is 1p23.
      (We could rewrite this as p2+0p+13 to match the desired form.)

      Solution 2:

      Let x be the probability that Bianca wins the tournament.
      There are three possible pairings for the first two matches:

      1. Bianca versus Alain, and Chen versus Dave

      2. Bianca versus Chen, and Alain versus Dave

      3. Bianca versus Dave, and Alain versus Chen

      Each of these three pairings occurs with probability 13.
      In (i), Bianca wins either if Bianca beats Alain, Chen beats Dave, and Bianca beats Chen, or if Bianca beats Alain, Dave beats Chen, and Bianca beats Dave.
      Since Bianca beats Alain with probability 12, Chen beats Dave with probability 1p, and Bianca beats Chen with probability 12, then the first possibility has probability 12(1p)12.
      Since Bianca beats Alain with probability 12, Dave beats Chen with probability p, and Bianca beats Dave with probability 1p, then the second possibility has probability 12p(1p).
      Therefore, the probability of Bianca winning, given that possibility (i) occurs, is 12(1p)12+12p(1p).
      In (ii), Bianca wins either if Bianca beats Chen, Alain beats Dave, and Bianca beats Alain, or if Bianca beats Alain, Dave beats Alain, and Bianca beats Dave.
      The combined probability of these is 12(1p)12+12p(1p).
      In (iii), Bianca wins either if Bianca beats Dave, Alain beats Chen, and Bianca beats Alain, or if Bianca beats Dave, Chen beats Alain, and Bianca beats Chen.
      The combined probability of these is (1p)1212+(1p)1212.
      Therefore, x=13(14(1p)+12p(1p)+14(1p)+12p(1p)+14(1p)+14(1p))=13(p(1p)+(1p))=13(pp2+1p) Thus, the probability that Bianca wins the tournament is 1p23.

    2. Throughout this solution, we will mostly not include units, but will assume that all lengths are in kilometres, all times are in seconds, and all speeds are in kilometres per second.
      We place the points in the coordinate plane with B at (0,0), A on the negative x-axis, and C on the positive x-axis.
      We put A at (1,0) and C at (2,0).
      Suppose that P has coordinates (x,y) and that the distance from P to B is d km.

      P(x, y) is located in the first quadrant.

      Since the sound arrives at A 12 s after arriving at B and sound travels at 13 km/s, then A is (12 s)(13 km/s)=16 km farther from P than B is.
      Thus, the distance from P to A is (d+16) km.
      Since the sound arrives at C an additional 1 second later, then C is an additional 13 km farther, and so is (d+16) km+(13 km)=(d+12) km from P.
      Since the distance from P to B is d km, then (x0)2+(y0)2=d2.
      Since the distance from P to A is (d+16) km, then (x+1)2+(y0)2=(d+16)2.
      Since the distance from P to C is (d+12) km, then (x2)2+(y0)2=(d+12)2.
      When these equations are expanded and simplified, we obtain x2+y2=d2x2+2x+1+y2=d2+13d+136x24x+4+y2=d2+d+14 Subtracting the first equation from the second, we obtain 2x+1=13d+136 Subtracting the first equation from the third, we obtain 4x+4=d+14 Therefore, 2(2x+1)+(4x+4)=2(13d+136)+(d+14)6=23d+118+d+14216=24d+2+36d+9(multiplying by 36)205=60dd=4112 Therefore, the distance from B to P is 4112 km.

    1. After each round, each L shape is divided into 4 smaller L shapes.
      This means that the number of L shapes increases by a factor of 4 after each round.
      After 1 round, there are 4 L shapes.
      After 2 rounds, there are 42=16 L’s of the smallest size.
      After 3 rounds, there are 43=64 L’s of the smallest size.

    2. There are four orientations of L shapes of a given size: A Bottom Left L: An L oriented so its bend is in the bottom left., A Top Right L: An L oriented so its bend is in the top right., A Bottom Right L: An L oriented so its bend is in the bottom right., A Top Left L: An L oriented so its bend is in the top left..

      When an L of each orientation is subdivided, the following figures are obtained:

      A larger Bottom Left L subdivided into 4 smaller L shapes. A smaller Bottom Left L makes up the bend in the bottom left with a smaller Bottom Right L to its left and a smaller Top Left L above it. Another smaller Bottom Left L completes the middle of the shape.� A larger Top Right L subdivided into 4 smaller L shapes. It is the subdivided Bottom Left L but rotated 180 degrees. A larger Bottom Right L subdivided into 4 smaller L shapes. It is the subdivided Bottom Left L but rotated 90 degrees counterclockwise. A larger Top Left L subdivided into 4 smaller L shapes. It is the subdivided Bottom Left L but rotated 90 degrees clockwise.

      From these figures, we can see that after each subsequent round,

      • Each Bottom Left L produces 2 Bottom Left L, 0 Top Right L, 1 Bottom Right L, and 1 Top Left L of the smallest size.

      • Each Top Right L produces 0 Bottom Left L, 2 Top Right L, 1 Bottom Right L, and 1 Top Left L.

      • Each Bottom Right L produces 1 Bottom Left L, 1 Top Right L, 2 Bottom Right L, and 0 Top Left L.

      • Each Top Left L produces 1 Bottom Left L, 1 Top Right L, 0 Bottom Right L, and 2 Top Left L.

      After 1 round, there are 2 Bottom Left L, 0 Top Right L, 1 Bottom Right L, and 1 Top Left L.
      After 2 rounds, the number of L’s of each orientation are as follows:

      • Bottom Left L : 22+00+11+11=6

      • Top Right L : 20+02+11+11=2

      • Bottom Right L : 21+01+12+10=4

      • Top Left L : 21+01+10+12=4

      After 3 rounds, the number of L’s of each orientation are as follows:

      • Bottom Left L : 62+20+41+41=20

      • Top Right L : 60+22+41+41=12

      • Bottom Right L : 61+21+42+40=16

      • Top Left L : 61+21+40+42=16

      Where do these numbers come from?
      For example, to determine the number of Bottom Left L after 2 rounds, we look at the number of L’s of each orientation after round 1 (2, 0, 1, 1) and ask how many Bottom Left L each of these produces at the next level. Since the four types each produce 2, 0, 1, and 1 Bottom Left L, then the total number of Bottom Left L after 2 rounds equals 22+00+11+11 which equals 6.
      As a second example, to determine the number of Top Right L after 3 rounds, we note that after 2 rounds the number of L’s of the four different orientations are 6, 2, 4, 4 and that each L of each of the four types produces 0, 2, 1, 1 Top Right L. This means that the total number of Top Right L after 3 rounds is 60+22+41+41=12.
      Putting all of this together, the number of L’s of the smallest size in the same orientation as the original L is 20.

    3. In (b), we determined the number of L’s of the smallest size in each orientation after 1, 2 and 3 rounds.
      We continue to determine the number of L’s of the smallest size after 4 rounds.
      After 4 rounds, the number of L’s of each orientation are as follows:

      • Bottom Left L : 202+120+161+161=72

      • Top Right L : 200+122+161+161=56

      • Bottom Right L : 201+121+162+160=64

      • Top Left L : 201+121+160+162=64

      This gives us the following tables of the numbers of L’s of the smallest size in each orientation after 1, 2, 3, and 4 rounds:

      After Round Bottom Left L Top Right L Bottom Right L Top Left L
      1 2 0 1 1
      2 6 2 4 4
      3 20 12 16 16
      4 72 56 64 64

      We re-write these numbers in the third row as 16+4,164,16,16 and the numbers in the fourth row as 64+8,648,64,64.
      Based on this, we might guess that the numbers of L’s of the smallest size in each orientation after n rounds are 4n1+2n1, 4n12n1, 4n1, 4n1.
      If this guess is correct, then, after 2020 rounds, the number of L’s of the smallest size in the same orientation as the original L is 42019+22019.
      We prove that these guesses are right by using an inductive process.
      First, we note that the table above shows that our guess is correct when n=1,2,3,4.
      Next, if we can show that our guess being correct after a given number of rounds implies that it is correct after the next round, then it will be correct after every round. This is because being correct after 4 rounds will mean that it is correct after 5 rounds, being correct after 5 rounds will mean that it is correct after 6 rounds, and so on to be correct after any number of rounds.
      Suppose, then, that after k rounds the numbers of L’s of the smallest size in each orientation are 4k1+2k1, 4k12k1, 4k1, 4k1.
      After k+1 rounds (that is, after the next round), the number of L’s of each orientation is:

      • Bottom Left L : (4k1+2k1)2+(4k12k1)0+4k11+4k11=44k1+22k1=4k+2k

      • Top Right L : (4k1+2k1)0+(4k12k1)2+4k11+4k11=44k122k1=4k2k

      • Bottom Right L : (4k1+2k1)1+(4k12k1)1+4k12+4k10=44k1=4k

      • Top Left L : (4k1+2k1)1+(4k12k1)1+4k10+4k12=44k1=4k

      Since k=(k+1)1, these expressions match our guess. This means that our guess is correct after every number of rounds.
      Therefore, after 2020 rounds, the number of L’s of the smallest size in the same orientation as the original L is 42019+22019.

    1. Here, the pairwise sums of the numbers a1a2a3a4 are s1s2s3s4s5s6.
      The six pairwise sums of the numbers in the list can be expressed as a1+a2,a1+a3,a1+a4,a2+a3,a2+a4,a3+a4 Since a1a2a3a4, then the smallest sum must be the sum of the two smallest numbers. Thus, s1=a1+a2.
      Similarly, the largest sum must be the sum of the two largest numbers, and so s6=a3+a4.
      Since a1a2a3a4, then the second smallest sum is a1+a3. This is because a1+a3 is no greater than each of the four sums a1+a4, a2+a3, a2+a4, and a3+a4:

      Since a3a4, then a1+a3a1+a4.
      Since a1a2, then a1+a3a2+a3.
      Since a1a2 and a3a4, then a1+a3a2+a4.
      Since a1a4, then a1+a3a3+a4.

      Thus, s2=a1+a3.
      Using a similar argument, s5=a2+a4.
      So far, we have s1=a1+a2 and s2=a1+a3 and s5=a2+a4 and s6=a3+a4.
      This means that s3 and s4 equal a1+a4 and a2+a3 in some order.
      It turns out that either order is possible.
      Case 1: s3=a1+a4 and s4=a2+a3
      Here, a1+a2=8 and a1+a3=104 and a2+a3=110.
      Adding these three equations gives (a1+a2)+(a1+a3)+(a2+a3)=8+104+110 and so 2a1+2a2+2a3=222 or a1+a2+a3=111.
      Since a2+a3=110, then a1=(a1+a2+a3)(a2+a3)=111110=1.
      Since a1=1 and a1+a2=8, then a2=7.
      Since a1=1 and a1+a3=104, then a3=103.
      Since a3=103 and a3+a4=208, then a4=105.
      Thus, (a1,a2,a3,a4)=(1,7,103,105).
      Case 2: s3=a2+a3 and s4=a1+a4
      Here, a1+a2=8 and a1+a3=104 and a2+a3=106.
      Using the same process, a1+a2+a3=109.
      From this, we obtain (a1,a2,a3,a4)=(3,5,101,107).
      Therefore, Kerry’s two possible lists are 1,7,103,105 and 3,5,101,107.

    2. Suppose that the values of s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 are fixed, but unknown.
      In terms of the numbers a1a2a3a4a5, the ten pairwise sums are a1+a2,a1+a3,a1+a4,a1+a5,a2+a3,a2+a4,a2+a5,a3+a4,a3+a5,a4+a5 These will equal s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 in some order.
      Using a similar analysis to that in (a), the smallest sum is a1+a2 and the largest sum is a4+a5. Thus, s1=a1+a2 and s10=s4+s5.
      Also, the second smallest sum will be s2=a1+a3 and the second largest sum will be s9=a3+a5.
      We let S=s1+s2+s3+s4+s5+s6+s7+s8+s9+s10 Note that S has a fixed, but unknown, value.
      Even though we do not know the order in which these pairwise sums are assigned to s1 through s10, the value of S will equal the sum of these ten pairwise expressions.
      In other words, S=4a1+4a2+4a3+4a4+4a5, since each of the numbers in the list occurs in four sums.
      Thus, a1+a2+a3+a4+a5=14S and so (a1+a2)+a3+(a4+a5)=14S.
      This means that s1+a3+s10=14S and so a3=14Ss1s10.
      Since the values of s1, s10 and S are fixed, then we are able to determine the value of a3 from the list of sums s1 through s10.
      Using the value of a3, the facts that s2=a1+a3 and s9=a3+a5, and that s2 and s9 are known, we can determine a1 and a5.
      Finally, using s1=a1+a2 and s10=a4+a5 and the values of a1 and a5, we can determine a2 and a4.
      Therefore, given the ten sums s1 through s10, we can determine the values of a3, a1, a5, a2, a4 and so there is only one possibility for the list a1,a2,a3,a4,a5. (Can you write out expressions for each of a1 through a5 in terms of s1 through s10 only?)

    3. Suppose that the lists a1, a2, a3, a4 and b1, b2, b3, b4 produce the same list of sums s1, s2, s3, s4, s5, s6. (Examples of such lists can be found in (a).)
      Let x be a positive integer. Consider the following list with 8 entries: a1,a2,a3,a4,b1+x,b2+x,b3+x,b4+x From this list, there are three categories of pairwise sums:

      1. ai+aj, 1i<j4: these give the sums s1 through s6

      2. (bi+x)+(bj+x), 1i<j4: each of these is 2x greater than the six sums s1 through s6 because the pairwise sums bi+bj give the six sums s1 through s6

      3. ai+(bj+x), 1i4 and 1j4

      Consider also the list with 8 entries: a1+x,a2+x,a3+x,a4+x,b1,b2,b3,b4 From this list, there are again three categories of pairwise sums:

      1. bi+bj, 1i<j4: these give the sums s1 through s6

      2. (ai+x)+(aj+x), 1i<j4: each of these is 2x greater than the six sums s1 through s6 because the pairwise sums ai+aj give the six sums s1 through s6

      3. (ai+x)+bj, 1i4 and 1j4

      Thus, the 28 pairwise sums in each case are the same. In each case, there are 6 sums in (i), 6 sums in (ii), and 16 sums in (iii).
      If we choose the initial lists to have the same pairwise sums and choose the value of x to be large enough so that ai+x is not equal to any bj and bi+x is not equal to any aj, we obtain two different lists of 8 numbers that each produce the same list of 28 sums.
      For example, if we choose a1,a2,a3,a4 to be 1,7,103,105 and b1,b2,b3,b4 to be 3,5,101,107 and x=10000, we get the lists 1,7,103,105,10003,10005,10101,10107 and 3,5,101,107,10001,10007,10103,10105 Using a similar analysis to that above, if the lists a1, a2, a3, a4, a5, a6, a7, a8 and b1, b2, b3, b4, b5, b6, b7, b8 have the same set of pairwise sums, then the lists a1,a2,a3,a4,a5,a6,a7,a8,b1+y,b2+y,b3+y,b4+y,b5+y,b6+y,b7+y,b8+y and a1+y,a2+y,a3+y,a4+y,a5+y,a6+y,a7+y,a8+y,b1,b2,b3,b4,b5,b6,b7,b8 will also have the same pairwise sums.
      Therefore, setting y=1000000, we see that the lists 1,7,103,105,10003,10005,10101,10107,1000003,1000005,1000101,1000107, 1010001,1010007,1010103,1010105 and 3,5,101,107,10001,10007,10103,10105,1000001,1000007,1000103,1000105, 1010003,1010005,1010101,1010107 have the same list of sums s1,s2,,s120, as required.