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2024 Euclid Contest
Solutions
(Grade 12)

Wednesday, April 3, 2024
(in North America and South America)

Thursday, April 4, 2024
(outside of North American and South America)

©2024 University of Waterloo


    1. For every x0, we note that x4+3x2x2=x2+3.
      Therefore, when x=2, we have x4+3x2x2=x2+3=22+3=7.
      Alternatively, when x=2, we have x4+3x2x2=24+32222=284=7.

    2. By the Pythagorean Theorem in ABC, we have AC2=AB2+BC2(t+1)2=102+(t1)2t2+2t+1=100+t22t+14t=100 and so t=25.
      Alternatively, we could remember the Pythagorean triple 5-12-13 and scale this triple by a factor of 2 to obtain the Pythagorean triple 10-24-26, noting that the difference between t+1 and t1 is 2 as is the difference between 26 and 24, which gives t+1=26 and so t=25.

    3. Since 2y+32y=14, then 42y+32y=14 or 72y=14.
      Therefore, 2y=714=12 and so y=14.

    1. Suppose that the sequence is a,b,c,13,e,36.
      Since 36=13+e, then e=3613=23.
      Since e=c+13, then 23=c+13 and so c=10.
      Since 13=b+c and c=10, then b=3.
      Since c=a+b and c=10 and b=3, then a=103=7.
      Therefore, the first term is 7.

    2. From the given information, we obtain 5r2+5r3=(5r)2 and so 5r2+5r3=25r2.
      Since r0, we can divide by 5r2 to obtain 1+r=5, which gives r=4.

    3. Suppose that Jimmy’s marks on his first, second, third, and fourth tests were w, x, y, and z, respectively.
      Since Jimmy’s average on his first, second and third tests was 65, then w+x+y3=65 or w+x+y=195.
      Since Jimmy’s average on his second, third and fourth tests was 80, then x+y+z3=80 or x+y+z=240.
      Since his mark on the fourth test was 2 times his mark on the first test, then z=2w.
      Thus, w+x+y=195 and x+y+2w=240.
      Subtracting the first equation from the second equation, we obtain w=45 and so his mark on the fourth test was z=2w=90.

    1. Since y=r(x3)(xr) passes through (0,48), then 48=r(03)(0r).
      Thus, 48=3r2 and so r2=16 or r=±4.

    2. With 13% sales tax on an item whose price is $B, the total cost is $(1.13B).
      With 5% sales tax on an item whose price is $B, the total cost is $(1.05B).
      From the given information $(1.13B)$(1.05B)=$24 or 1.13B1.05B=24.
      Therefore, 0.08B=24, which gives B=300.
      Alternatively, we could note that the difference in total prices is the difference in the amount of tax paid. This is the difference between 13% of the original price and 5% of the original price; this difference is equal to 8% of the original price. If 8% of the original price is equal to $24, then 1% of the original price is equal to $3 and so the original price is $3×100=$300.

    3. When n=1, f(2n)=(f(n))2 becomes f(2)=(f(1))2.
      Since f(1)=3, then f(2)=32=9.
      When m=1, f(2m+1)=3f(2m) becomes f(3)=3f(2).
      Since f(2)=9, then f(3)=39=27.
      When n=2, f(2n)=(f(n))2 becomes f(4)=(f(2))2.
      Since f(2)=9, then f(4)=92=81.
      Therefore, f(2)+f(3)+f(4)=9+27+81=117.

    1. Since ABD is right-angled at B and has ADB=45°, then BAD=45°.
      Similarly, CPD is right-angled and isosceles with PCD=45°.
      Further, APN and CBN are also both right-angled and isosceles.
      Since CB=6 and NB=CB, then NB=6.
      Since AB=12 and NB=6, then AN=ABNB=6.

      Since APN is right-angled and isosceles, then its sides are in the ratio 1:1:2.
      Thus, AP=PN=12AN=62=32.
      Alternatively, if AP=PN=x, then the Pythagorean Theorem gives AN2=AP2+PN2 and so 62=2x2 which gives AP2=x2=18.
      Thus, the area of APN is 12APPN=123232=9.

    2. The line with equation y=3x+6 has y-intercept 6, which means that OB=6.
      To find the x-intercept of this line, we set y=0 and obtain the equation 3x+6=0 which gives 3x=6 or x=2. This means that OA=2.
      Since ABO is right-angled at O, its area is 12OBOA=1262=6.
      Since the area of ACD is 12 of the area of ABO, then the area of ACD is 3.
      Next, we note that the line with equation y=mx+1 has y-intercept 1; thus, OD=1.
      This means that the area of ADO is 12ODOA=1212=1.

      We can determine the area of BCD by subtracting the areas of ACD and ADO from that of ABO, which tells us that the area of BCD is 631=2.

      Now, we can consider BD, which has length 61=5, as the base of BCD; the corresponding height of BCD is the distance from C to the y-axis, which we call h.
      Thus, 125h=2 and so h=45.
      This means that C has x-coordinate 45.
      Since C is on the line with equation y=3x+6, we have y=345+6=185.
      Therefore, the coordinates of C are (45,185).

    1. Suppose that AP=w, PD=x, AS=y, and SB=z.

      We use the notation |APXS| to represent the area of APXS, and so on.
      Thus, |APXS|=wy, |PDRX|=xy, |SXBQ|=wz, and |XRCQ|=xz.
      Then, |APXS||XRCQ|=wyxz=xywz=|PDRX||SXQB| If |APXS|=2, |PDRX|=3, and |SWQB|=6, then a=|XRQC|=263=4.
      If |APXS|=2, |PDRX|=6, and |SWQB|=3, then a=|XRQC|=236=1.
      If |APXS|=6, |PDRX|=2, and |SWQB|=3, then a=|XRQC|=632=9.
      Since we are told that there are three possible values for a, then these are 1, 4 and 9.
      (Can you explain why there are exactly three such values?)

    2. The x-intercepts of the parabola with equation y=x24tx+5t26t are x=4t±(4t)24(5t26t)2 The distance, d, between these intercepts is their difference, which is d=4t+(4t)24(5t26t)24t(4t)24(5t26t)2=(4t)24(5t26t) From this we see that d is as large as possible exactly when the discriminant is as large as possible. Here, the discriminant, Δ, is Δ=(4t)24(5t26t)=16t220t2+24t=4t2+24t Completing the square, Δ=4(t26t)=4(t26t+99)=4(t26t+9)+36=4(t3)2+36 Since (t3)20, then Δ36 and Δ=36 exactly when (t3)2=0 or t=3.
      Therefore, the discriminant is maximized when t=3, which means that the distance between the x-intercepts is as large as possible when t=3.

    1. Every multiple of 21 is of the form 21k for some integer k.
      For such a multiple to be between 10 000 and 100 000, we need 10000<21k<100000 or 1000021<k<10000021.
      Since 1000021476.2 and 100000214761.9 and k is an integer, then 477k4761. (Note that k is greater than 476.2 and is an integer, so must be at least 477; similarly, k is at most 4761.)
      We also want the units digit of 21k to be 1.
      This means that the units digit of k itself is 1, since the units digit of the product of 21 and k is equal to units digit of k because the units digit of 21 is 1.
      Therefore, the possible values of k are 481,491,501,,4751,4761.
      There are 429 such values. To see this, we can see that counting the integers in this list is the same as counting the integers in the list 48,49,50,,475,476. This list is equivalent to removing the integers from 1 to 47 from the list of integers from 1 to 476, giving 47647=429 integers.
      Thus, M=429.

    2. Solution 1:

      We can partition the N students at Strickland S.S. into four groups:

      • a students who are in the physics club and are in the math club

      • b students who are in the physics club and are not in the math club

      • c students who are not in the physics club but are in the math club

      • d students who are not in the physics club and are not in the math club

      In Math Club Not in Math Club
      In Physics Club a b
      Not in Physics Club c d

      From the given information, there are 25N students in the physics club. In other words, a+b=25N.
      Among the students in the physics club, twice as many are not in the math club as are in the math club. This means that b=2325N=415N and a=1325N=215N.
      From the given information, there are 14N students in the math club. In other words, a+c=14N.
      Since a=215N, then c=14N215N=760N.
      Since a+b+c+d=N, then d=Nabc=N415N215N760N=2960N.
      Lastly, we know that 500<N<600.
      Since each of a, b, c, and d is an integer, then N must be divisible by 60.
      Therefore, N=540 and so the number of students not in either club is d=2960540=261.

      Solution 2:

      Since there are N students at Strickland S.S., then 25N are in the physics club and 14N are in the math club.
      Since each of 25N and 14N must be an integer, then N must be divisible by 5 and must be divisible by 4.
      Since 5 and 4 share no common divisor larger than 1, then N must be divisible by 54=20.
      Thus, we let N=20m for some positive integer m.
      In this case, 25N=8m students are in the physics club and 14N=5m students are in the math club.
      Now, among the 8m students in the physics club, twice as many are not in the math club as are in the math club.
      In other words, 13 of the 8m students in the physics club are in the math club. This means that m must be divisible by 3, since 3 is a prime number and 8 is not divisible by 3.
      Therefore, m=3k for some positive integer k, which means that N=20m=60k and 25N=8m=24k and 14N=5m=15k.
      Since 500<N<600 and N is a multiple of 60, then N=540, which means that k=9.
      Thus, the number of students in the physics club is 24k=216, of whom 13216=72 are in the math club and 23216=144 are not in the math club.
      Also, the number of students in the math club is 15k=135.
      Finally, we know that

      • there are 540 students at the school,

      • 72 of whom are in both the physics club and the math club,

      • 144 of whom are in the physics club and not in the math club, and

      • 13572=63 are in the math club and not in the physics club.

      Therefore, the number of students in neither club is 5407214463=261.

    1. Suppose that the length of the track is 2L m, that Arun’s constant speed is a m/s, and that Bella’s constant speed is b m/s.
      When Arun and Bella run over the same interval of time, the ratio of the distances that they run is equal to the ratio of their speeds.
      Consider the interval of time from the start to when they first meet. In the diagram, A is Arun’s starting point, B is Bella’s starting point, and P is this first meeting point.

      Since Arun has run 100 m and together they have covered half of the length of the track, then Bella has run (L100) m.
      Thus, ab=100L100.
      From their first meeting point P to their second meeting point, which we label Q, Bella runs 150 m.

      Over this time, Arun runs from P to B to Q.
      Since Bella runs 150100=50 m past A, then QB=(L50) m (because AB=L m and AQ=50 m) and so Arun runs (L100) m+(L50) m which is equal to (2L150) m.
      Thus, over this second interval of time, ab=2L150150.
      Equating expressions for ab and solving, 100L100=2L150150100150=(L100)(2L150)15000=2L2350L+15000350L=2L2 Since L0, then 2L=350, and so the total length of the track is 350 m.

      Checking, if the length of the track is 350 m, then half of the length is 75 m.
      This means that from the start to P, Arun runs 100 m and Bella runs 75 m.
      Also, from P to Q, Bella runs 150 m and Arun runs 200 m.
      Note that 10075=200150 so these numbers are consistent with the given information.

    2. Using exponent laws, the following equations are equivalent: 41+cos3θ=22cosθ8cos2θ(22)1+cos3θ=22cosθ(23)cos2θ22+2cos3θ=22cosθ23cos2θ22+2cos3θ=22cosθ+3cos2θ2+2cos3θ=2cosθ+3cos2θ2cos3θ3cos2θ+cosθ=0cosθ(2cos2θ3cosθ+1)=0cosθ(2cosθ1)(cosθ1)=0 and so cosθ=0 or cosθ=1 or cosθ=12.
      Since 0°θ360°, the solutions are θ=90°,270°,0°,360°,60°,300°.
      Listing these in increasing order, the solutions to the original equation are θ=0°,60°,90°,270°,300°,360°

    1. We join B to E and A to D.
      Since MC is tangent to the circles with centres A and B at D and E, respectively, then AD and BE are perpendicular to MC.
      Since the radius of the circle with centre B is 3, then AB=3 and BE=3.
      Since the radius of the circle with centre A is 4, then AD=4 and AT=4.
      Let CB=x and MT=y.

      We note that CEB, CDA and CTM are all similar, since they are right-angled at E, D and T, respectively, and share a common angle at C.
      Since CEB and CDA are similar, then CBCA=BEAD and so xx+3=34 which gives 4x=3x+9 and so x=9.
      By the Pythagorean Theorem, CE=CB2BE2=9232=72=62.
      Since CEB and CTM are similar, then BECE=MTCT and so 362=y9+3+4 which gives y=16362=82=42.

      Finally, the area of MNC is equal to 12MNCT.
      If we joined B to G, we would see that CEB is congruent to CGB (each is right-angled, they have a common hypotenuse, and BE=BG). This means that BCE=BCG, which in turn means that MT=TN.
      Since MT=TN, then MN=242=82 and so the area of MNC is 128216 or 642.

    2. First, we note that log3z=log10zlog103=2log10z2log103=log10(z2)log10(32)=log10(z2)log109=log9(z2) Similarly, log4y=log16(y2) and log5x=log25(x2).
      We also note from the original system of equations that x>0 and y>0 and z>0.
      Therefore, we can re-write the original system of equations as log9x+log9y+log9(z2)=2log16x+log16(y2)+log16z=1log25(x2)+log25y+log25z=0 Using logarithm rules, this is equivalent to the system log9(xyz2)=2log16(xy2z)=1log25(x2yz)=0 and to the system xyz2=92=81xy2z=161=16x2yz=250=1 Multiplying these three equations together, we obtain x4y4z4=1296 and so (xyz)4=64.
      Thus, xyz=6.
      Since xyz2=81 and xyz=6, then z=xyz2xyz=816=272.
      Similarly, y=xy2zxyz=166=83 and x=x2yzxyz=16.
      Therefore, (x,y,z)=(16,83,272).
      We can check by substitution that this triple does satisfy the original system of equations.

    1. Suppose that a sequence of n steps includes p steps in the positive direction and m steps in the negative direction.
      This means that n=p+m (total number of steps) and d=pm (final position).
      Since n=9 and d=5, then p+m=9 and pm=5 which give p=7 and m=2.
      Thus, we need to count the number of sequences of 9 steps that include 7 steps in the positive direction and 2 steps in the negative direction.
      This is the same as creating a sequence of 9 letters, 7 of which are R and 2 of which are L.
      There are (92)=982=36 such sequences.
      (Alternatively, we could see that there are 36 sequences by counting directly: there are 8 sequences where the first L appears in the first position, 7 where the first L appears in the second position, and so on.)

    2. Solution 1:

      Since n=9 and d=3, then p+m=9 and pm=3 which gives p=6 and m=3.
      This means that we can think about sequences of 9 letters that include 6 R’s and 3 L’s.
      In this context, a direction change happens when the sequence changes from a block of one letter to a block of another letter (that is, when the sequence has an occurrence of "RL" or "LR").
      For the sequence to change directions an even number of times, the number of blocks of letters in the sequence is odd.
      For the number of blocks to be odd, the sequence must begin and end with the same letter.
      If the sequence of 9 letters begins and ends with an R, the 7 letters in between include 4 R’s and 3 L’s; there are (73)=765321=35 such sequences.
      If the sequence of 9 letters begins and ends with an L, the 7 letters in between include 6 R’s and 1 L; there are (71)=7 such sequences.
      In total, there are 35+7=42 such sequences.

      Solution 2:

      As in Solution 1, we want to count the number of sequences of 9 letters that include 6 R’s and 3 L’s and that have an odd number of blocks.
      Since there are 3 L’s, there can be at most 3 blocks of L’s, and so at most 7 blocks.

      Case 1: There are 3 blocks

      Suppose that the block order is R-L-R.
      Here, there are 3 L’s in the block of L’s and 6 R’s distributed between two blocks of R’s.
      There are 5 ways to distribute the R’s: 1+5, 2+4, 3+3, 4+2, 5+1.
      This means that there are 5 such sequences.
      Suppose that the block order is L-R-L.
      The block of R’s includes 6 R’s and there are 3 L’s to distribute between two blocks.
      There are 2 ways to do this: 1+2 or 2+1.
      This means that there are 2 such sequences.
      In total, there are 5+2=7 sequences in this case.

      Case 2: There are 7 blocks

      Since there cannot be 4 blocks of L’s, the block order must be R-L-R-L-R-L-R.
      Each block of L’s includes exactly 1 L, and there are 2 additional R’s to distribute after one R is placed in each block.
      There are 4 ways to put these two R’s in the same block, and 6 ways to put them in separate blocks (1st and 2nd, 1st and 3rd, 1st and 4th, 2nd and 3rd, 2nd and 4th, 3rd and 4th).
      Therefore, there are 10 different sequences in this case.

      Case 3: There are 5 blocks

      Suppose that the block order is R-L-R-L-R.
      The two blocks of L’s include 3 L’s in total; there are 2 ways to distribute these (1+2 or 2+1).
      The three blocks of L’s include 6 R’s in total.
      Starting with one R in each block, there are 3 additional R’s to distribute.
      There are 3 ways in which the 3 R’s can go in the same block: 3+0+0, 0+3+0, 0+0+3.
      There are 6 ways in which the 3 R’s can go in two blocks: 2+1+0, 2+0+1, 1+2+0, 0+2+1, 1+0+2, 0+1+2.
      There is 1 way in which the 3 R’s can go in three different blocks: 1+1+1.
      Thus, there are 2 ways to distribute the L’s and 3+6+1=10 ways to distribute the R’s and so 210=20 such sequences.
      Suppose that the block order is L-R-L-R-L.
      Each block of L’s includes exactly 1 L, and the two blocks of R’s include exactly 6 R’s.
      There are 5 ways to distribute these R’s, as we saw in a similar situation in Case 1.
      This means that there are 5 such sequences.
      In total, there are 20+5=25 sequences in this case.

      Combining the cases, there are 10+7+25=42 such sequences.

    3. Consider the sequences of length n that end at x=d with d0.
      Suppose that such a sequence includes p steps in the positive direction and m steps in the negative direction.
      Since p+m=n (total number of steps) and pm=d (ending position), then 2p=n+d (which gives p=n+d2) and 2m=nd (which gives m=nd2).
      This means that all sequences of length n that end at x=d with d0 correspond to the same values of p and m, and so we treat p and m as fixed in what follows.
      Before proceeding to the general case, we deal with some specific small values.

      When n=1, there is exactly 1 sequence that ends with d0. This sequence moves 1 step to the right. Since there is an odd number of sequences when n=1, it cannot be the case that half of the sequences have an even number of changes of direction.

      When n=2, we can either have p=2 and m=0 (giving d=2) or p=1 and m=1 (giving d=0).
      When n=2 and d=2, there is only 1 sequence, and so it cannot be the case that half of the sequences in this category have an even number of changes of direction.
      When n=2 and d=0, there are 2 sequences (RL or LR), each of which has 1 change of direction, so it is not the case that half of the sequences in this category have an even number of changes of direction.

      Therefore, we assume that n3. Since n=p+m and pm, then p2 as well.
      We think about sequences of n letters that include p R’s and m L’s.
      As in (b) Solution 1, the number of direction changes is even exactly when the number of blocks of letters is odd, which is exactly when the sequence begins and ends with the same letter.
      For such a sequence to begin and end with R, the n2 letters between include (p2) R’s and m L’s. (Note that n20 and p20 since n2 and p0.)
      For such a sequence to begin and end with L, the n2 letters between include p R’s and (m2) L’s. (Note that n20. It is possible, though, that m2<0, in which case we adopt the convention that there are 0 such sequences. In this case, we have p=nm=(n2)(m2)>n2 and so (n2p)=0, making the calculations that follow consistent with this convention.)
      Thus, there are (n2p2)+(n2p) such sequences.
      This is true for exactly half of all sequences when the following equivalent equations are true: (n2p2)+(n2p)=12(np)(n2)!(p2)!m!+(n2)!p!(m2)!=n!2p!m!(n2)!(p2)!m!+(n2)!p!(m2)!=n(n1)(n2)!2p!m!1(p2)!m!+1p!(m2)!=n(n1)2p!m!p!(p2)!+m!(m2)!=n(n1)22p(p1)+2m(m1)=n(n1)2p2+2m2=n2n+2p+2m(p2+2mp+m2)+(p22mp+m2)=n2(p+m)+2p+2m(p+m)2+(pm)2=n2+(p+m)n2+d2=n2+nd2=n Therefore, exactly half of these sequences have an even number of changes of direction exactly when n=d2.
      To finish the problem, we now need to count the number of perfect squares (that is, possible values for n) in the correct range.
      Since 2n2024 and d0 and n is a perfect square, then the facts that 12=1, 22=4, 442=1936, and 452=2025 tell us that there are 43 perfect squares from 2 to 2024, inclusive, and so 43 such pairs (d,n). These are the pairs (d,d2) for d=2,3,,43,44.

    1. Here is the general set-up for this problem along with the specific instance in (a) where s=1:

      Triangle ABC with A at (negative 1, 0), C at (1,0), and B at (0,4). T is at (negative t, 0), between A and the origin, and S is at (s,0), between the origin and C. P is on AB and Q is on BC and PS and QT cross at point X inside triangle ABC.    Triangle ABC with A at (negative 1, 0), C at (1,0), and B at (0,4). T is at (negative t, 0), between A and the origin, and S is at (1,0), coinciding with C. P is on AB and Q is on BC and PS and QT cross at point X inside triangle ABC.

      Since SP and TQ divide ABC into four regions of equal area, then APS, which is made up of two of these regions, has area equal to one-half of the area of ABC.
      Since S and C coincide, then P is the midpoint of AB, which means that P has coordinates (12,2).

      Alternatively, we could note that the area of ABC is 1224=4 and so the area of APS must be 2.
      If P has y-coordinate p, then 122p=2 which gives p=2.
      Since AB has slope 4 and y-intercept 4, its equation is y=4x+4.
      Since P lies on AB and has y-coordinate 2, its x-coordinate satisfies 2=4x+4 and so 4x=2 or x=12.
      Thus, the coordinates of P are (12,2).

    2. We note that the area of ABC is 1224=4. When the triangle is divided into four equal areas, each of these areas must be 1.
      Suppose that (s,t) is a balancing pair.
      This is true exactly when

      • the area of SXT is 1, and

      • the area of APS is 2, and

      • the area of CQT is 2.

      (Bullets 1 and 2 tell us that the area of quadrilateral APXT is 1.
      Bullets 1 and 3 tell us that the area of quadrilateral CQXS is 1.
      Since three of the areas are 1, then the fourth must be 1.)

      SXT has base TS=s+t. If its height is h, then 12(s+t)h=1 and so h=2s+t.
      APS has base AS=1+s. If its height is p, then 12(s+1)p=2 and so p=4s+1.
      CQT has base TC=1+t. If its height is q, then 12(t+1)q=2 and so q=4t+1.
      The line through A and B has slope 4 and y-intercept 4, so its equation is y=4x+4.
      The line through C and B has slope 4 and y-intercept 4, so its equation is y=4x+4.
      Since P lies on the line with equation y=4x+4 and the y-coordinate of P is 4s+1, then the x-coordinate of P satisfies 4s+1=4x+4 which gives 1s+1=x+1, from which we obtain x=1s+11=1s1s+1=ss+1.
      Therefore, P has coordinates (ss+1,4s+1).
      Since Q lies on the line with equation y=4x+4 and the y-coordinate of Q is 4t+1, then the x-coordinate of Q satisfies 4t+1=4x+4 which gives 1t+1=x+1 or x=11t+1=t+11t+1=tt+1.
      Therefore, Q has coordinates (tt+1,4t+1).
      Next, we find the coordinates of X by finding the equations of the lines through P and S, and through Q and T.
      The slope of the line through S and P is 4s+10ss+1s=4ss(s+1)=4s2+2s Since this line passes through S(s,0), its equation is y=4s2+2s(xs).
      The slope of the line through T and Q is 4t+10tt+1(t)=4t+t(t+1)=4t2+2t Since this line passes through S(t,0), its equation is y=4t2+2t(x+t).
      To find the x-coordinate of X, we find the point of intersection of the line through S and P and the line through T and Q; thus, we solve 4s2+2s(xs)=4t2+2t(x+t)4ss2+2s4tt2+2t=(4s2+2s+4t2+2t)x4s(t2+2t)4t(s2+2s)=(4(t2+2t)+4(s2+2s))xx=st2s2tt2+2t+s2+2s Therefore, the y-coordinate of X is obtained from y=4t2+2t(st2s2tt2+2t+s2+2s+t)=4t2+2tst2s2t+t3+2t2+ts2+2stt2+2t+s2+2s=4t2+2tt3+2t2+st2+2stt2+2t+s2+2s=4t2+2tt(t2+2t)+s(t2+2t)t2+2t+s2+2s=4(s+t)t2+2t+s2+2s Finally, the y-coordinate of X is equal to the height h from earlier, so 2s+t=4(s+t)t2+2t+s2+2s1s+t=2(s+t)t2+2t+s2+2st2+2t+s2+2s=2(s+t)2t2+2t+s2+2s=2s2+4st+2t24st+2s+2t=s2+t2 Therefore, the desired relationship is true, with d=4, e=f=2 and g=0.

    3. We look for pairs of the form (s,t)=(s,ks) where s and k are rational numbers.
      From the relationship from (b), s2+k2s2=4ks2+2s+2kss2(k2+4k+1)=(2k+2)ss(k2+4k+1)=2k+2(since s>0)s=2k+2k2+4k+1 Thus, t=ks=2k2+2kk2+4k+1.
      Since k>0, then s>0.
      We need to have st. This is equivalent to 2k+2k2+4k+12k2+2kk2+4k+12k+22k2+2k(since k2+4k+1>0)22k21k(since k>0) Thus, when k1, we have 0<st.
      Finally, we need t1. This is equivalent to 2k2+2kk2+4k+112k2+2kk2+4k+1(since k2+4k+1>0)k22k10 The roots of k22k1=0 are k=2±(2)24(1)(1)2=1±2.
      Thus, k22k10 when 12k1+2.
      Since k>0, then t1 when 0<k1+2.
      Therefore, when 1k1+2, we have 0<st1.
      We note that there are infinitely many rational numbers in any interval of non-zero length; in particular, there are infinitely many rational numbers k that satisfy 1k1+2.
      We do need to confirm that the values of 2k+2k2+4k+1 are different for different values of k, which will confirm that there are infinitely many different values of s as k assumes these infinitely many values itself.
      To do this, we show that if 2k+2k2+4k+1=2j+2j2+4j+1, then k=j; this fact will allow us to conclude that if kj, then 2k+2k2+4k+12j+2j2+4j+1.
      When k>0 and j>0, then following equations are equivalent: 2k+2k2+4k+1=2j+2j2+4j+1(2k+2)(j2+4j+1)=(2j+2)(k2+4k+1)(k+1)(j2+4j+1)=(j+1)(k2+4k+1)j2k+4jk+k+j2+4j+1=jk2+4jk+j+k2+4k+1j2kjk2+3j3k+j2k2=0(jk)(jk+j+k+3)=0 Since jk+j+k+3>0, then it must be the case that jk=0 and so j=k.
      This means that if kj, then 2k+2k2+4k+12j+2j2+4j+1 and so there are infinitely many different values of s.
      Therefore, the pairs (s,t)=(2k+2k2+4k+1,2k2+2kk2+4k+1) where k is a rational number with 1k1+2 is an infinite family of solutions to the relationship from (b).