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2021 Euclid Contest
Solutions

Wednesday, April 7, 2021
(in North America and South America)

Thursday, April 8, 2021
(outside of North American and South America)

©2021 University of Waterloo


    1. Since (a1)+(2a3)=14, then 3a=18 and so a=6.

    2. Since (c2c)+(2c3)=9, then c2+c3=9 and so c2+c12=0.

      Factoring, we obtain (c+4)(c3)=0 and so c=3 or c=4.

    3. Solution 1

      Manipulating algebraically, we obtain the following equivalent equations: 1x2+32x2=102+3=20x2(multiplying through by 2x2, given that x0)5=20x2x2=14 and so x=±12.

      Solution 2

      Manipulating algebraically, we obtain the following equivalent equations: 1x2+32x2=1022x2+32x2=1052x2=105=20x2(since x0)x2=14 and so x=±12.

    1. Using a calculator, we see that (103+1)2=10012=1002001 The sum of the digits of this integer is 1+2+1 which equals 4.

      To determine this integer without using a calculator, we can let x=103.

      Then (103+1)2=(x+1)2=x2+2x+1=(103)2+2(103)+1=1002001

    2. Before the price increase, the total cost of 2 small cookies and 1 large cookie is 2$1.50+$2.00=$5.00.

      10% of $1.50 is 0.1$1.50=$0.15. After the price increase, 1 small cookie costs $1.50+$0.15=$1.65.

      5% of $2.00 is 0.05$2.00=$0.10. After the price increase, 1 large cookie costs $2.00+$0.10=$2.10.

      After the price increase, the total cost of 2 small cookies and 1 large cookie is 2$1.65+$2.10=$5.40.

      The percentage increase in the total cost is $5.40$5.00$5.00×100%=40500×100%=8%.

    3. Suppose that Rayna’s age is x years.

      Since Qing is twice as old as Rayna, Qing’s age is 2x years.

      Since Qing is 4 years younger than Paolo, Paolo’s age is 2x+4 years.

      Since the average of their ages is 13 years, we obtain x+(2x)+(2x+4)3=13 This gives 5x+4=39 and so 5x=35 or x=7.

      Therefore, Rayna is 7 years old, Qing is 14 years old, and Paolo is 18 years old.

      (Checking, the average of 7, 14 and 18 is 7+14+183=393=13.)

    1. The length of PQ is equal to (05)2+(120)2=(5)2+122=13.

      In a similar way, we can see that QR=RS=SP=13.

      Therefore, the perimeter of PQRS is 413=52.

      (We can also see that if O is the origin, then POQ, POS, ROQ, and ROS are congruent because OQ=OS and OP=OR, which means that PQ=QR=RS=SP.)

    2. Solution 1

      Suppose that B has coordinates (r,s) and C has coordinates (t,u).

      Since M(3,9) is the midpoint of A(0,8) and B(r,s), then 3 is the average of 0 and r (which gives r=6) and 9 is the average of 8 and s (which gives s=10).

      Since N(7,6) is the midpoint of B(6,10) and C(t,u), then 7 is the average of 6 and t (which gives t=8) and 6 is the average of 10 and u (which gives u=2).

      The slope of the line segment joining A(0,8) and C(8,2) is 8208 which equals 34.

      Solution 2

      Since M is the midpoint of AB and N is the midpoint of BC, then MN is parallel to AC.

      Therefore, the slope of AC equals the slope of the line segment joining M(3,9) to N(7,6), which is 9637 or 34.

    3. Since V(1,18) is on the parabola, then 18=2(12)+4(1)+c and so c=18+24=16.

      Thus, the equation of the parabola is y=2x2+4x+16.

      The y-intercept occurs when x=0, and so y=16. Thus, D has coordinates (0,16).

      The x-intercepts occur when y=0. Here, 2x2+4x+16=02(x22x8)=02(x4)(x+2)=0 and so x=4 and x=2.

      This means that E and F, in some order, have coordinates (4,0) and (2,0).

      Therefore, DEF has base EF of length 4(2)=6 and height 16 (vertical distance from the x-axis to the point D).

      Finally, the area of DEF is 12616=48.

    1. We obtain successively 3(8x)+5(8x)=2618(8x)=2618x+1=261(23)x+1=26123(x+1)=261 Thus, 3(x+1)=61 and so 3x+3=61 which gives 3x=58 or x=583.

    2. Since the list 3n2, m2, 2(n+1)2 consists of three consecutive integers written in increasing order, then 2(n+1)23n2=22n2+4n+23n2=2n2+4n=0n(n4)=0 and so n=0 or n=4.

      If n=0, the list becomes 0, m2, 2. This means that m2=1 and so m=±1.

      If n=4, we have 3n2=316=48 and 2(n+1)2=225=50 giving the list 48, m2, 50. This means that m2=49 and so m=±7.

      Thus, the possible values for m are 1, 1, 7, 7.

    1. Solution 1

      Suppose that S0 has coordinates (a,b).

      Step 1 moves (a,b) to (a,b).

      Step 2 moves (a,b) to (a,b+2).

      Step 3 moves (a,b+2) to (a,b+2).

      Thus, S1 has coordinates (a,b+2).

      Step 1 moves (a,b+2) to (a,b2).

      Step 2 moves (a,b2) to (a,b).

      Step 3 moves (a,b) to (a,b).

      Thus, S2 has coordinates (a,b), which are the same coordinates as S0.

      Continuing this process, S4 will have the same coordinates as S2 (and thus as S0) and S6 will have the same coordinates as S4, S2 and S0.

      Since the coordinates of S6 are (7,1), the coordinates of S0 are also (7,1).

      Solution 2

      We work backwards from S6(7,1).

      To do this, we undo the Steps of the process P by applying them in reverse order.

      Since Step 3 reflects a point in the y-axis, its inverse does the same.

      Since Step 2 translates a point 2 units upwards, its inverse translates a point 2 units downwards.

      Since Step 1 reflects a point in the x-axis, its inverse does the same.

      Applying these inverse steps to S6(7,1), we obtain (7,1), then (7,3), then (7,3).

      Thus, S5 has coordinates (7,3).

      Applying the inverse steps to S5(7,3), we obtain (7,3), then (7,1), then (7,1).

      Thus, S4 has coordinates (7,1), which are the same coordinates as S6.

      If we apply these steps two more times, we will see that S2 is the same point as S4.

      Two more applications tell us that S0 is the same point as S2.

      Therefore, the coordinates of S0 are the same as the coordinates of S6, which are (7,1).

    2. We begin by determining the length of AB in terms of x.

      Since ABDE is a rectangle, BD=AE=2x.

      Since BCD is equilateral, DBC=60.

      Join A to D.

      Diagonal AD is drawn in rectangle ABCD. Triangle BCD is beside rectangle ABDE joined at common side BD.

      Since AD and BC are parallel, ADB=DBC=60.

      Consider ADB. This is a 30-60-90 triangle since ABD is a right angle.

      Using ratios of side lengths, ABBD=31 and so AB=3BD=23x, which is the answer to (i).

      Next, we determine ACAD.

      Now, ADBD=21 and so AD=2BD=4x.

      Suppose that M is the midpoint of AE and N is the midpoint of BD.

      Since AE=BD=2x, then AM=ME=BN=ND=x.

      Join M to N and N to C and A to C.

      Since ABDE is a rectangle, then MN is parallel to AB and so MN is perpendicular to both AE and BD.

      Also, MN=AB=23x.

      Since BCD is equilateral, its median CN is perpendicular to BD.

      Since MN and NC are perpendicular to BD, MNC is actually a straight line segment and so MC=MN+NC.

      Now BNC is also a 30-60-90 triangle, and so NC=3BN=3x.

      This means that MC=23x+3x=33x.

      Finally, AMC is right-angled at M and so AC=AM2+MC2=x2+(33x)2=x2+27x2=28x2=27x since x>0.

      This means that ACAD=27x4x=72=74, which means that the integers r=7 and s=4 satisfy the conditions for (ii).

    1. Solution 1

      Since the sequence t1,t2,t3,,tn2,tn1,tn is arithmetic, then t1+tn=t2+tn1=t3+tn2 This is because, if d is the common difference, we have t2=t1+d and tn1=tnd, as well as having t3=t1+2d and tn2=tn2d.

      Since the sum of all n terms is 1000, using one formula for the sum of an arithmetic sequence gives n2(t1+tn)=1000n(t1+tn)=2000n(t3+tn2)=2000n(5+95)=2000 and so n=20.

      Solution 2

      Suppose that the arithmetic sequence with n terms has first term a and common difference d.

      Then t3=a+2d=5 and tn2=a+(n3)d=95.

      Since the sum of the n terms equals 1000, then n2(2a+(n1)d)=1000 Adding the equations a+2d=5 and a+(n3)d=95, we obtain 2a+(n1)d=100.

      Substituting, we get n2(100)=1000 from which we obtain n=20.

    2. Since the sum of a geometric sequence with first term a, common ratio r and 4 terms is 6+62, then a+ar+ar2+ar3=6+62 Since the sum of a geometric sequence with first term a, common ratio r and 8 terms is 30+302, then a+ar+ar2+ar3+ar4+ar5+ar6+ar7=30+302 But a+ar+ar2+ar3+ar4+ar5+ar6+ar7=(a+ar+ar2+ar3)+r4(a+ar+ar2+ar3)=(1+r4)(a+ar+ar2+ar3) Therefore, 30+302=(1+r4)(6+62)30+3026+62=1+r45=1+r4r4=4r2=2(since r2>0)r=±2 If r=2, a+ar+ar2+ar3=a+2a+a(2)2+a(2)3=a+2a+2a+22a=a(3+32) Since a+ar+ar2+ar3=6+62, then a(3+32)=6+62 and so a=6+623+32=2.

      If r=2, a+ar+ar2+ar3=a2a+a(2)2+a(2)3=a2a+2a22a=a(332) Since a+ar+ar2+ar3=6+62, then a(332)=6+62 and so a=6+62332=2+2212=(2+22)(1+2)(12)(1+2)=2+22+22+412=642 Therefore, the possible values of a are a=2 and a=642.

      An alternate way of arriving at the equation 1+r4=5 is to use the formula for the sum of a geometric sequence twice to obtain a(1r4)1r=6+62a(1r8)1r=30+302 assuming that r1. (Can you explain why r1 and r41 without knowing already that r=±2?)

      Dividing the second equation by the first, we obtain a(1r8)1r1ra(1r4)=30+3026+62 which gives 1r81r4=5 Since 1r8=(1+r4)(1r4), we obtain 1+r4=5. We then can proceed as above.

    1. Victor stops when there are either 2 green balls on the table or 2 red balls on the table.

      If the first 2 balls that Victor removes are the same colour, Victor will stop.

      If the first 2 balls that Victor removes are different colours, Victor does not yet stop, but when he removes a third ball, its colour must match the colour of one of the first 2 balls and so Victor does stop.

      Therefore, the probability that he stops with at least 1 red ball and 1 green ball on the table is equal to the probability that the first 2 balls that he removes are different colours.

      Also, the probability that the first 2 balls that he removes are different colours is equal to 1 minus the probability that the first 2 balls that he removes are the same colour.

      The probability that the first two balls that Victor draws are both green is 3726 because for the first ball there are 7 balls in the bag, 3 of which are green and for the second ball there are 6 balls in the bag, 2 of which are green.

      The probability that the first two balls that Victor draws are both red is 4736 because for the first ball there are 7 balls in the bag, 4 of which are red and for the second ball there are 6 balls in the bag, 3 of which are red.

      Thus, the probability that the first two balls that Victor removes are the same colour is 3726+4736=17+27=37 This means that the desired probability is 137=47.

    2. Using the definition of f, the following equations are equivalent: f(a)=02a23a+1=0(a1)(2a1)=0 Therefore, f(a)=0 exactly when a=1 or a=12.

      Thus, f(g(sinθ))=0 exactly when g(sinθ)=1 or g(sinθ)=12.

      Using the definition of g,

      • g(b)=1 exactly when log12b=1, which gives b=(12)1=12, and

      • g(b)=1/2 exactly when log12b=1/2, which gives b=(12)1/2=12.

      Therefore, f(g(sinθ))=0 exactly when sinθ=12 or sinθ=12.

      Since 0θ2π, the solutions are θ=16π,56π,14π,34π.

    1. Suppose that the integers in the first row are, in order, a, b, c, d, e.

      Using these, we calculate the integer in each of the boxes below the top row in terms of these variables, using the rule that each integer is the product of the integers in the two boxes above:

      Five rows of expressions, with five expressions in the top row, followed by 4, 3, 2, and 1 in the next four rows in order. The expressions in each row are given in the following list.

      Therefore, ab4c6d4e=9953280000.

      Next, we determine the prime factorization of the integer 9953280000: 9953280000=104995328=245423124416=27542315552=21054231944=2135423243=2165435=2163554 Thus, ab4c6d4e=2163554.

      Since the right side is not divisible by 7, none of a, b, c, d, e can equal 7.

      Thus, a, b, c, d, e are five distinct integers chosen from {1,2,3,4,5,6,8}.

      The only one of these integers divisible by 5 is 5 itself.

      Since 2163554 includes exactly 4 factors of 5, then either b=5 or d=5. No other placement of the 5 can give exactly 4 factors of 5.

      Case 1: b=5

      Here, ac6d4e=21635 and a, c, d, e are four distinct integers chosen from {1,2,3,4,6,8}.

      Since ac6d4e includes exactly 5 factors of 3 and the possible values of a, c, d, e that are divisible by 3 are 3 and 6, then either d=3 and one of a and e is 6, or d=6 and one of a and e is 3. No other placements of the multiples of 3 can give exactly 5 factors of 3.

      Case 1a: b=5, d=3, a=6

      Here, ac6d4e=6c634e=235c6e.

      This gives c6e=215 and c and e are distinct integers from {1,2,4,8}.

      Trying the four possible values of c shows that c=4 and e=8 is the only solution in this case. Here, (a,b,c,d,e)=(6,5,4,3,8).

      Case 1b: b=5, d=3, e=6 We obtain (a,b,c,d,e)=(8,5,4,3,6).

      Case 1c: b=5, d=6, a=3

      Here, ac6d4e=3c664e=2435c6e.

      This gives c6e=212 and c and e are distinct integers from {1,2,4,8}.

      Trying the four possible values of c shows that c=4 and e=1 is the only solution in this case. Here, (a,b,c,d,e)=(3,5,4,6,1).

      Case 1d: b=5, d=6, e=3 We obtain (a,b,c,d,e)=(1,5,4,6,3).

      Case 2: d=5: A similar analysis leads to 4 further quintuples (a,b,c,d,e).

      Therefore, there are 8 ways in which the integers can be chosen and placed in the top row to obtain the desired integer in the bottom box.

    2. Let N=(1!)(2!)(3!)(398!)(399!)(400!)200!.

      For each integer k from 1 to 200, inclusive, we rewrite (2k)! as 2k(2k1)!.

      Therefore, (2k1)!(2k)!=(2k1)!2k(2k1)!=2k((2k1)!)2.

      (In particular, (1!)(2!)=2(1!)2, (3!)(4!)=4(3!)2, and so on.)

      Thus, N=2(1!)24(3!)2398(397!)2400(399!)2200! Re-arranging the numerator of the expression, we obtain N=(1!)2(3!)2(397!)2(399!)2(24398400)200! We can now re-write 24398400 as (21)(22)(2199)(2200).

      Since there are 200 sets of parentheses, we obtain N=(1!)2(3!)2(397!)2(399!)22200(12199200)200! Since 12199200=200!, we can conclude that N=2200(1!)2(3!)2(397!)2(399!)2 Therefore, N=2100(1!)(3!)(397!)(399!) which is a product of integers and thus an integer itself.

      Since N is an integer, N is a perfect square, as required.

    1. When a=5 and b=4, we obtain a2+b2ab=52+4254=21.

      Therefore, we want to find all pairs of integers (K,L) with K2+3L2=21.

      If L=0, then L2=0, which gives K2=21 which has no integer solutions.

      If L=±1, then L2=1, which gives K2=18 which has no integer solutions.

      If L=±2, then L2=4, which gives K2=9 which gives K=±3.

      If L=±3, then L2=9. Since 3L2=27>21, then there are no real solutions for K.

      Similarly, if L2>9, there are no real solutions for K.

      Therefore, the solutions are (K,L)=(3,2),(3,2),(3,2),(3,2).

    2. Suppose that K and L are integers.

      Then (K+L)2+(KL)2(K+L)(KL)=(K2+2KL+L2)+(K22KL+L2)(K2L2)=K2+3L2 Therefore, the integers a=K+L and b=KL satisfy the equation K2+3L2=a2+b2ab, and so for all integers K and L, there is at least one pair of integers (a,b) that satisfy the equation.

      How could we come up with this? One way to do this would be trying some small values of K and L, calculating K2+3L2 and using this to make a guess, which can then be proven algebraically as above. In particular, here are some values:

      K L K2+3L2=a2+b2ab a b
      1 1 4 2 0
      2 1 7 3 1
      3 1 12 4 2
      1 2 13 3 -1
      2 2 16 4 0
      3 2 21 5 1
      The columns for a and b might lead us to guess that a=K+L and b=KL, which we proved above does in fact work.

    3. Suppose that a and b are integers.

      If a is even, then a2 is an integer and so (a2b)2+3(a2)2=a242a2b+b2+3a24=a2+b2ab Thus, if K=a2b and L=a2, we have K2+3L2=a2+b2ab.

      If b is even, then b2 is an integer and so a similar algebraic argument shows that (b2a)2+3(b2)2=a2+b2ab and so if K=b2a and L=b2, we have K2+3L2=a2+b2ab.

      If a and b are both odd, then a+b and ab are both even, which means that a+b2 and ab2 are both integers, and so (a+b2)2+3(ab2)2=a2+2ab+b24+3a26ab+3b24=4a2+4b24ab4=a2+b2ab Thus, if K=a+b2 and L=ab2, we have K2+3L2=a2+b2ab.

      Therefore, in all cases, for all integers a and b, there is at least one pair of integers (K,L) with K2+3L2=a2+b2ab.

      As in (b), trying some small cases might help us make a guess of possible expressions for K and L in terms of a and b:

      a b K2+3L2=a2+b2ab K L
      1 1 1 1 0
      2 1 3 0 1
      3 1 7 2 1
      4 1 13 1 2
      1 2 3 0 1
      2 2 4 1 1
      3 2 7 2 1
      4 2 12 3 1
      5 3 19 4 1
      While there might not initially seem to be useful patterns here, re-arranging the rows and adding some duplicates might help show a pattern:
      a b K2+3L2=a2+b2ab K L
      2 1 3 0 1
      4 1 13 1 2
      2 2 4 1 1
      4 2 12 3 1
      1 2 3 0 1
      3 2 7 2 1
      4 2 12 3 1
      1 1 1 1 0
      3 1 7 2 1
      5 3 19 4 1

    1. We label the centres of the outer circles, starting with the circle labelled Z and proceeding clockwise, as A, B, C, D, E, F, G, H, J, and K, and the centre of the circle labelled Y as L.

      Join L to each of A, B, C, D, E, F, G, H, J, and K. Join A to B, B to C, C to D, D to E, E to F, F to G, G to H, H to J, J to K, and K to A.

      When two circles are tangent, the distance between their centres equals the sum of their radii.

      Thus, BC=CD=DE=EF=FG=GH=HJ=JK=2+1=3BL=DL=FL=HL=KL=2+4=6CL=EL=GL=JL=1+4=5AB=AK=r+2AL=r+4 By side-side-side congruence, the following triangles are congruent: BLC,DLC,DLE,FLE,FLG,HLG,HLJ,KLJ Similarly, ALB and ALK are congruent by side-side-side.

      Let ALB=θ and let BLC=α.

      By congruent triangles, ALK=θ and BLC=DLC=DLE=FLE=FLG=HLG=HLJ=KLJ=α The angles around L add to 360 and so 2θ+8α=360 which gives θ+4α=180 and so θ=1804α.

      Since θ=1804α, then cosθ=cos(1804α)=cos4α.

      Consider ALB and BLC.

      Triangle ABL and triangle BLC are side by side, joined at common side BL. AB has length r+2, AL has length r+4, BL has length 6, CL has length 5, and BC has length 3. Angle ALB is theta and angle BLC is alpha.

      By the cosine law in ALB, AB2=AL2+BL22ALBLcosθ(r+2)2=(r+4)2+622(r+4)(6)cosθ12(r+4)cosθ=r2+8r+16+36r24r4cosθ=4r+4812(r+4)cosθ=r+123r+12 By the cosine law in BLC, BC2=BL2+CL22BLCLcosα32=62+522(6)(5)cosα60cosα=36+259cosα=5260cosα=1315 Since cosα=1315, then cos2α=2cos2α1=21692251=338225225225=113225 and cos4α=2cos22α1=2113222521=25538506255062550625=2508750625 Finally, cosθ=cos4αr+123r+12=2508750625r+12r+4=2508716875(r+4)+8r+4=25087168751+8r+4=25087168758r+4=8212168752r+4=205316875r+42=168752053r+4=337502053r=255382053 Therefore, the positive integers s=25538 and t=2053 satisfy the required conditions.

    2. Let the centre of the middle circle be O, and the centres of the other circles be P, Q, R, and S, as shown.

      Join O to P, Q, R, and S, and join P to Q, Q to R, R to S, and S to P.

      The circle in the middle has radius c and centre O. The circles with radius a have centres P and R, and the circles with radius b have centres Q and S. Sides PQ, QR, RS, and SP are made up of two segments: one of length a and one of length b. Segments OP and OR are each made up of two segments: one of length a and one of length c. OS and QS are each made up of two segments: one of length b and one of length c.

      Using a similar argument as in (a), we see that OP=OR=a+cOQ=OS=b+cPQ=QR=RS=SP=a+b By side-side-side congruence, OPQ, OPS, ORQ, and ORS are congruent.

      This means that POQ=POS=ROQ=ROS.

      Since POQ+POS+ROQ+ROS=360 (these angles surround O), then POQ=14360=90 This means that OPQ is right-angled at O.

      By the Pythagorean Theorem, PQ2=OP2+OQ2 and so (a+b)2=(a+c)2+(b+c)2.

      Manipulating algebraically, the following equations are equivalent: (a+b)2=(a+c)2+(b+c)2a2+2ab+b2=a2+2ac+c2+b2+2bc+c22ab=2ac+2bc+2c2ab=ac+bc+c2abacbc=c2abacbc+c2=2c2a(bc)c(bc)=2c2(ac)(bc)=2c2 Therefore, if a, b and c are real numbers for which the diagram can be constructed, then a, b and c satisfy this last equation.

      Also, if real numbers a, b and c satisfy the final equation, then (a+b)2=(a+c)2+(b+c)2 (because these equations were equivalent) and so the triangle with side lengths a+b, a+c and b+c is right-angled with hypotenuse a+b (because the Pythagorean Theorem works in both directions), which means that four such triangles can be assembled to form a rhombus PQRS with side lengths a+b and centre O, which means that the five circles can be drawn by marking off the appropriate lengths a, b and c and drawing the circles as in the original diagram.

      In other words, the diagram can be drawn exactly when (ac)(bc)=2c2.

      Suppose that c is a fixed positive integer.

      Determining the value of f(c) is thus equivalent to counting the number of pairs of positive integers (a,b) with c<a<b and (ac)(bc)=2c2.

      Since a and b are integers with a>c and b>c, the integers ac and bc are positive and form a positive divisor pair of the integer 2c2.

      Since a<b, we have ac<bc and so ac and bc are distinct integers.

      Also, since c>0, 2c2=2c which is not an integer since c is an integer, which means that 2c2 is not a perfect square.

      Therefore, every pair (a,b) corresponds to a positive divisor pair of 2c2 (namely, ac and bc).

      Similarly, every divisor pair e and g of 2c2 with e>g gives a pair of positive integers (a,b) with a<b by setting a=e+c and b=g+c.

      In other words, f(c) is exactly the number of positive divisor pairs of 2c2. (Again, we note that 2c2 is not a perfect square.)

      Therefore, we want to determine all positive integers c for which the integer 2c2 has an even number of divisor pairs, which means that we want to determine all positive integers c for which the number of positive divisors of 2c2 is a multiple of 4 (because each positive divisor pair corresponds to 2 positive divisors and 2 times an even integer is a multiple of 4).

      Suppose that the prime factorization of c is c=2rp1e1p2e2pkek for some integer k0, integer r0, odd prime numbers p1, p2, , pk, and positive integers e1, e2, , ek.

      Then 2c2=22r+1p12e1p22e2pk2ek and so 2c2 has (2r+2)(2e1+1)(2e2+1)(2ek+1) positive divisors.

      The first factor in this product is even and each factor after the first is odd.

      Therefore, this product is a multiple of 4 exactly when 2r+2 is a multiple of 4.

      This is true exactly when 2r+2=4s for some positive integer s and so 2r=4s2 or r=2s1.

      In other words, the number of positive divisors of 2c2 is a multiple of 4 exactly when r is an odd integer.

      Finally, this means that the positive integers for which f(c) are even are exactly those positive integers that have exactly an odd number of factors of 2 in their prime factorization.