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A thin triangle (Posted on 2021-03-11) Difficulty: 3 of 5
ΔABnC is isosceles with vertex A.

B1, B2, ..., Bn-1 are unique points alternating between the legs of the triangle where

AB1 = B1B2 = B2B3 = ... = Bn-1Bn = BnC.

If ∠CABn = 0.8°, find n.

No Solution Yet Submitted by Jer    
Rating: 3.0000 (1 votes)

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Solution computer solution | Comment 2 of 3 |
Let the length of AB1 be the unit of length. The base is of length 1. Since the vertex at A has an angular measure of 0.8°, the height of the skinny triangle is .5/tan(0.4°) ~= 71.6185608347376, taking one side (one half) of the isosceles triangle as a right triangle.

Segment AB1 is the only one that lies along one side of the triangle. After that the 1-unit segments span from side to side of the triangle, forming small isosceles triangles along the way, with their bases lying along one side or the other of the larger triangle, alternating sides.

The first such smaller triangle, the only one that has a side along one of the sides of the large triangle, has base angles of 0.8°, as one of them is coincident with the apex of the large triangle. Thus its vertex angle is 180 - 2*.8 = 178.4°. Using the law of cosines, its base is sqrt(1+1-2*cos(178.4°)) ~= 1.99980504801861. That is the first distance measured down along one side of the large triangle, from A to B2, with B1 having been at the apex of this first triangle. The height attained thus far being cos(0.4°) times the slant height, with the slant height being the accumulated bases of the small isosceles triangles on just one side or the other in alternate values of k along the way to n.

From then on, each small isosceles triangle faces the opposite way. The first time only, the new base angle is the supplement of the preceding apex angle; each time after that the new base angle is 180 minus the previous apex angle and minus the second previous base angle. For example, in the triangle that leads to B4 in the table below, the base angle of 2.4° is 180 - 176.8 - 0.8. The new apex angle is of course 180 minus twice the base angle.

The table below eventually reaches B112 at the full height of the large triangle, so n = 112. Incidentally since the even B's are on the opposite side from B1, that first segment lay on AC.

It's interesting that a rational number of degrees would lead to such a situation. Offhand I'd say it indicates it is more related to the circumference of a circle than to the radius or diameter. It creates a suspicion that it might not be exact and the zeros after the integral portion are misleading, but I've added more about that further down; the result is verified. 

  k   base  apex    base      slant      height  relative
     angle  angle             height     so far   height
  
  2  0.800 178.400 1.99981  1.9998050  1.9997563 0.0279223 
  3  1.600 176.800 1.99922  2.9992202  2.9991471 0.0418767 
  4  2.400 175.200 1.99825  3.9980507  3.9979533 0.0558229 
  5  3.200 173.600 1.99688  4.9961018  4.9959800 0.0697582 
  6  4.000 172.000 1.99513  5.9931788  5.9930328 0.0836799 
  7  4.800 170.400 1.99299  6.9890875  6.9889172 0.0975853 
  8  5.600 168.800 1.99045  7.9836336  7.9834391 0.1114716 
  9  6.400 167.200 1.98754  8.9766233  8.9764046 0.1253363 
 10  7.200 165.600 1.98423  9.9678630  9.9676201 0.1391765 
 11  8.000 164.000 1.98054 10.9571595 10.9568924 0.1529896 
 12  8.800 162.400 1.97646 11.9443198 11.9440287 0.1667728 
 13  9.600 160.800 1.97199 12.9291515 12.9288365 0.1805235 
 14 10.400 159.200 1.96714 13.9114627 13.9111237 0.1942391 
 15 11.200 157.600 1.96191 14.8910618 14.8906990 0.2079168 
 16 12.000 156.000 1.95630 15.8677579 15.8673712 0.2215539 
 17 12.800 154.400 1.95030 16.8413605 16.8409501 0.2351478 
 18 13.600 152.800 1.94392 17.8116799 17.8112459 0.2486959 
 19 14.400 151.200 1.93717 18.7785269 18.7780692 0.2621956 
 20 15.200 149.600 1.93003 19.7417129 19.7412318 0.2756441 
 21 16.000 148.000 1.92252 20.7010503 20.7005458 0.2890388 
 22 16.800 146.400 1.91464 21.6563519 21.6558242 0.3023773 
 23 17.600 144.800 1.90638 22.6074316 22.6068807 0.3156567 
 24 18.400 143.200 1.89775 23.5541039 23.5535299 0.3288747 
 25 19.200 141.600 1.88875 24.4961843 24.4955874 0.3420285 
 26 20.000 140.000 1.87939 25.4334892 25.4328694 0.3551156 
 27 20.800 138.400 1.86965 26.3658357 26.3651932 0.3681335 
 28 21.600 136.800 1.85955 27.2930421 27.2923770 0.3810797 
 29 22.400 135.200 1.84909 28.2149278 28.2142402 0.3939515 
 30 23.200 133.600 1.83827 29.1313128 29.1306029 0.4067466 
 31 24.000 132.000 1.82709 30.0420187 30.0412866 0.4194623 
 32 24.800 130.400 1.81555 30.9468678 30.9461136 0.4320963 
 33 25.600 128.800 1.80367 31.8456837 31.8449077 0.4446460 
 34 26.400 127.200 1.79142 32.7382913 32.7374935 0.4571091 
 35 27.200 125.600 1.77883 33.6245165 33.6236971 0.4694830 
 36 28.000 124.000 1.76590 34.5041865 34.5033456 0.4817654 
 37 28.800 122.400 1.75261 35.3771298 35.3762677 0.4939539 
 38 29.600 120.800 1.73899 36.2431763 36.2422931 0.5060461 
 39 30.400 119.200 1.72503 37.1021572 37.1012530 0.5180396 
 40 31.200 117.600 1.71073 37.9539049 37.9529800 0.5299322 
 41 32.000 116.000 1.69610 38.7982534 38.7973079 0.5417214 
 42 32.800 114.400 1.68113 39.6350381 39.6340722 0.5534050 
 43 33.600 112.800 1.66584 40.4640958 40.4631098 0.5649808 
 44 34.400 111.200 1.65023 41.2852651 41.2842590 0.5764464 
 45 35.200 109.600 1.63429 42.0983856 42.0973597 0.5877996 
 46 36.000 108.000 1.61803 42.9032991 42.9022535 0.5990382 
 47 36.800 106.400 1.60146 43.6998484 43.6987834 0.6101600 
 48 37.600 104.800 1.58458 44.4878783 44.4867942 0.6211629 
 49 38.400 103.200 1.56739 45.2672353 45.2661322 0.6320447 
 50 39.200 101.600 1.54989 46.0377673 46.0366454 0.6428033 
 51 40.000 100.000 1.53209 46.7993242 46.7981837 0.6534365 
 52 40.800  98.400 1.51399 47.5517574 47.5505986 0.6639424 
 53 41.600  96.800 1.49560 48.2949204 48.2937435 0.6743188 
 54 42.400  95.200 1.47691 49.0286681 49.0274733 0.6845638 
 55 43.200  93.600 1.45794 49.7528576 49.7516452 0.6946753 
 56 44.000  92.000 1.43868 50.4673477 50.4661179 0.7046514 
 57 44.800  90.400 1.41914 51.1719991 51.1707521 0.7144901 
 58 45.600  88.800 1.39933 51.8666744 51.8654104 0.7241895 
 59 46.400  87.200 1.37924 52.5512382 52.5499575 0.7337477 
 60 47.200  85.600 1.35888 53.2255570 53.2242599 0.7431629 
 61 48.000  84.000 1.33826 53.8894994 53.8881861 0.7524332 
 62 48.800  82.400 1.31738 54.5429359 54.5416067 0.7615569 
 63 49.600  80.800 1.29624 55.1857392 55.1843944 0.7705320 
 64 50.400  79.200 1.27485 55.8177839 55.8164237 0.7793570 
 65 51.200  77.600 1.25321 56.4389468 56.4375714 0.7880300 
 66 52.000  76.000 1.23132 57.0491068 57.0477166 0.7965493 
 67 52.800  74.400 1.20920 57.6481450 57.6467402 0.8049134 
 68 53.600  72.800 1.18684 58.2359446 58.2345255 0.8131206 
 69 54.400  71.200 1.16425 58.8123910 58.8109578 0.8211692 
 70 55.200  69.600 1.14143 59.3773718 59.3759248 0.8290578 
 71 56.000  68.000 1.11839 59.9307768 59.9293163 0.8367847 
 72 56.800  66.400 1.09513 60.4724982 60.4710245 0.8443485 
 73 57.600  64.800 1.07165 61.0024304 61.0009438 0.8517477 
 74 58.400  63.200 1.04797 61.5204700 61.5189708 0.8589808 
 75 59.200  61.600 1.02409 62.0265161 62.0250046 0.8660465 
 76 60.000  60.000 1.00000 62.5204700 62.5189464 0.8729434 
 77 60.800  58.400 0.97572 63.0022354 63.0007001 0.8796700 
 78 61.600  56.800 0.95125 63.4717184 63.4701717 0.8862252 
 79 62.400  55.200 0.92659 63.9288275 63.9272696 0.8926076 
 80 63.200  53.600 0.90176 64.3734735 64.3719048 0.8988159 
 81 64.000  52.000 0.87674 64.8055698 64.8039905 0.9048491 
 82 64.800  50.400 0.85156 65.2250321 65.2234426 0.9107059 
 83 65.600  48.800 0.82621 65.6317787 65.6301793 0.9163851 
 84 66.400  47.200 0.80070 66.0257302 66.0241212 0.9218856 
 85 67.200  45.600 0.77503 66.4068098 66.4051915 0.9272064 
 86 68.000  44.000 0.74921 66.7749434 66.7733161 0.9323465 
 87 68.800  42.400 0.72325 67.1300590 67.1284231 0.9373048 
 88 69.600  40.800 0.69714 67.4720874 67.4704432 0.9420804 
 89 70.400  39.200 0.67090 67.8009621 67.7993098 0.9466723 
 90 71.200  37.600 0.64453 68.1166188 68.1149589 0.9510797 
 91 72.000  36.000 0.61803 68.4189961 68.4173288 0.9553016 
 92 72.800  34.400 0.59142 68.7080349 68.7063606 0.9593374 
 93 73.600  32.800 0.56468 68.9836790 68.9819979 0.9631860 
 94 74.400  31.200 0.53784 69.2458746 69.2441871 0.9668469 
 95 75.200  29.600 0.51089 69.4945705 69.4928770 0.9703194 
 96 76.000  28.000 0.48384 69.7297184 69.7280191 0.9736026 
 97 76.800  26.400 0.45670 69.9512723 69.9495676 0.9766961 
 98 77.600  24.800 0.42947 70.1591890 70.1574793 0.9795991 
 99 78.400  23.200 0.40216 70.3534281 70.3517136 0.9823112 
100 79.200  21.600 0.37476 70.5339517 70.5322328 0.9848318 
101 80.000  20.000 0.34730 70.7007245 70.6990015 0.9871603 
102 80.800  18.400 0.31976 70.8537140 70.8519874 0.9892964 
103 81.600  16.800 0.29217 70.9928905 70.9911605 0.9912397 
104 82.400  15.200 0.26451 71.1182268 71.1164937 0.9929897 
105 83.200  13.600 0.23681 71.2296985 71.2279626 0.9945461 
106 84.000  12.000 0.20906 71.3272837 71.3255455 0.9959087 
107 84.800  10.400 0.18127 71.4109636 71.4092234 0.9970771 
108 85.600   8.800 0.15344 71.4807218 71.4789799 0.9980511 
109 86.400   7.200 0.12558 71.5365447 71.5348014 0.9988305 
110 87.200   5.600 0.09770 71.5784213 71.5766770 0.9994152 
111 88.000   4.000 0.06980 71.6063437 71.6045987 0.9998050 
112 88.800   2.400 0.04188 71.6203062 71.6185608 1.0000000 

clc
CAB=.8;
halfCAB=CAB/2;
halfBase=.5;
height=halfBase/tand(halfCAB);
baseAngle(1)=CAB;
slant(1)=1; slant(2)=0;
currHeight=cosd(halfCAB);
side=2;
n=1;
while currHeight<=height
    vAngle=180-2*baseAngle(n);
    base=sqrt(2-2*cosd(vAngle));
    slant(side)=slant(side)+base;
    currHeight=slant(side)*cosd(halfCAB);
    n=n+1;
    fprintf('%3d %6.3f %7.3f %7.5f %10.7f %10.7f %9.7f '  ,...
        n,baseAngle(n-1),vAngle,base,slant(side),currHeight,currHeight/height)
    side=3-side;
    if n==2
        baseAngle(n)=180-vAngle;
    else
        baseAngle(n)=180-baseAngle(n-2)-vAngle; 
    end
end


Being that I was suspicious of the setup as involving a rational number of degrees, I wanted to be sure the result was not a fluke of rounding error giving a ratio of exactly 1. (In fact the ratio of large triangle height to the height at which the trellis ended was actually 1.00000000000002, making me feel justified that the 2 at the end was just a rounding error.)


So I redid it with vpa, variable precision arithmetic:


clc
digits(50)
CAB=vpa(.8);
halfCAB=CAB/2;
halfBase=vpa(.5);
height=halfBase/vpa(tand(halfCAB));
baseAngle(1)=CAB;
slant(1)=vpa(1); slant(2)=vpa(0);
currHeight=vpa(cosd(halfCAB));
side=vpa(2);
n=1;
while currHeight<=height
    vAngle=180-2*baseAngle(n);
    base=sqrt(2-2*vpa(cosd(vAngle)));
    slant(side)=slant(side)+base;
    currHeight=slant(side)*vpa(cosd(halfCAB));
    n=n+1;
    fprintf('%3d %6.3f %7.3f %7.5f %10.7f %10.7f %9.7f '  ,...
        n,baseAngle(n-1),vAngle,base,slant(side),currHeight,currHeight/height)
    if currHeight/height>.9999
       disp([currHeight currHeight/height])  % check on full accuracy
    end
    side=3-side;
    if n==2
        baseAngle(n)=180-vAngle;
    else
        baseAngle(n)=180-baseAngle(n-2)-vAngle; 
    end
end
disp([currHeight currHeight/height])  


The results were exact to 50 places, the first 32 of which (the actual height) agreed with the 32 that are afforded by default when using vpa. The 112th iteration does exactly match the height of the skinny triangle.

An artifact of doing it over is that the ratio did not exceed 1 immediately, as B112 to B113 was in fact a horizontal line and n could be either B112 or B113 depending on which one you want to call C instead. After that, in reality the points would go back up the thin triangle, but the trigonometry assumed positive length bases of the small triangles and kept increasong the height, stopping the iterations.

  k   base  apex    base      slant      height  relative
     angle  angle             height     so far   height
  
  2  0.800 178.400 1.99981  1.9998050  1.9997563 0.0279223 
  3  1.600 176.800 1.99922  2.9992202  2.9991471 0.0418767 
  4  2.400 175.200 1.99825  3.9980507  3.9979533 0.0558229 
  5  3.200 173.600 1.99688  4.9961018  4.9959800 0.0697582 
  6  4.000 172.000 1.99513  5.9931788  5.9930328 0.0836799 
  7  4.800 170.400 1.99299  6.9890875  6.9889172 0.0975853 
  8  5.600 168.800 1.99045  7.9836336  7.9834391 0.1114716 
  9  6.400 167.200 1.98754  8.9766233  8.9764046 0.1253363 
 10  7.200 165.600 1.98423  9.9678630  9.9676201 0.1391765 
 11  8.000 164.000 1.98054 10.9571595 10.9568924 0.1529896 
 12  8.800 162.400 1.97646 11.9443198 11.9440287 0.1667728 
 13  9.600 160.800 1.97199 12.9291515 12.9288365 0.1805235 
 14 10.400 159.200 1.96714 13.9114627 13.9111237 0.1942391 
 15 11.200 157.600 1.96191 14.8910618 14.8906990 0.2079168 
 16 12.000 156.000 1.95630 15.8677579 15.8673712 0.2215539 
 17 12.800 154.400 1.95030 16.8413605 16.8409501 0.2351478 
 18 13.600 152.800 1.94392 17.8116799 17.8112459 0.2486959 
 19 14.400 151.200 1.93717 18.7785269 18.7780692 0.2621956 
 20 15.200 149.600 1.93003 19.7417129 19.7412318 0.2756441 
 21 16.000 148.000 1.92252 20.7010503 20.7005458 0.2890388 
 22 16.800 146.400 1.91464 21.6563519 21.6558242 0.3023773 
 23 17.600 144.800 1.90638 22.6074316 22.6068807 0.3156567 
 24 18.400 143.200 1.89775 23.5541039 23.5535299 0.3288747 
 25 19.200 141.600 1.88875 24.4961843 24.4955874 0.3420285 
 26 20.000 140.000 1.87939 25.4334892 25.4328694 0.3551156 
 27 20.800 138.400 1.86965 26.3658357 26.3651932 0.3681335 
 28 21.600 136.800 1.85955 27.2930421 27.2923770 0.3810797 
 29 22.400 135.200 1.84909 28.2149278 28.2142402 0.3939515 
 30 23.200 133.600 1.83827 29.1313128 29.1306029 0.4067466 
 31 24.000 132.000 1.82709 30.0420187 30.0412866 0.4194623 
 32 24.800 130.400 1.81555 30.9468678 30.9461136 0.4320963 
 33 25.600 128.800 1.80367 31.8456837 31.8449077 0.4446460 
 34 26.400 127.200 1.79142 32.7382913 32.7374935 0.4571091 
 35 27.200 125.600 1.77883 33.6245165 33.6236971 0.4694830 
 36 28.000 124.000 1.76590 34.5041865 34.5033456 0.4817654 
 37 28.800 122.400 1.75261 35.3771298 35.3762677 0.4939539 
 38 29.600 120.800 1.73899 36.2431763 36.2422931 0.5060461 
 39 30.400 119.200 1.72503 37.1021572 37.1012530 0.5180396 
 40 31.200 117.600 1.71073 37.9539049 37.9529800 0.5299322 
 41 32.000 116.000 1.69610 38.7982534 38.7973079 0.5417214 
 42 32.800 114.400 1.68113 39.6350381 39.6340722 0.5534050 
 43 33.600 112.800 1.66584 40.4640958 40.4631098 0.5649808 
 44 34.400 111.200 1.65023 41.2852651 41.2842590 0.5764464 
 45 35.200 109.600 1.63429 42.0983856 42.0973597 0.5877996 
 46 36.000 108.000 1.61803 42.9032991 42.9022535 0.5990382 
 47 36.800 106.400 1.60146 43.6998484 43.6987834 0.6101600 
 48 37.600 104.800 1.58458 44.4878783 44.4867942 0.6211629 
 49 38.400 103.200 1.56739 45.2672353 45.2661322 0.6320447 
 50 39.200 101.600 1.54989 46.0377673 46.0366454 0.6428033 
 51 40.000 100.000 1.53209 46.7993242 46.7981837 0.6534365 
 52 40.800  98.400 1.51399 47.5517574 47.5505986 0.6639424 
 53 41.600  96.800 1.49560 48.2949204 48.2937435 0.6743188 
 54 42.400  95.200 1.47691 49.0286681 49.0274733 0.6845638 
 55 43.200  93.600 1.45794 49.7528576 49.7516452 0.6946753 
 56 44.000  92.000 1.43868 50.4673477 50.4661179 0.7046514 
 57 44.800  90.400 1.41914 51.1719991 51.1707521 0.7144901 
 58 45.600  88.800 1.39933 51.8666744 51.8654104 0.7241895 
 59 46.400  87.200 1.37924 52.5512382 52.5499575 0.7337477 
 60 47.200  85.600 1.35888 53.2255570 53.2242599 0.7431629 
 61 48.000  84.000 1.33826 53.8894994 53.8881861 0.7524332 
 62 48.800  82.400 1.31738 54.5429359 54.5416067 0.7615569 
 63 49.600  80.800 1.29624 55.1857392 55.1843944 0.7705320 
 64 50.400  79.200 1.27485 55.8177839 55.8164237 0.7793570 
 65 51.200  77.600 1.25321 56.4389468 56.4375714 0.7880300 
 66 52.000  76.000 1.23132 57.0491068 57.0477166 0.7965493 
 67 52.800  74.400 1.20920 57.6481450 57.6467402 0.8049134 
 68 53.600  72.800 1.18684 58.2359446 58.2345255 0.8131206 
 69 54.400  71.200 1.16425 58.8123910 58.8109578 0.8211692 
 70 55.200  69.600 1.14143 59.3773718 59.3759248 0.8290578 
 71 56.000  68.000 1.11839 59.9307768 59.9293163 0.8367847 
 72 56.800  66.400 1.09513 60.4724982 60.4710245 0.8443485 
 73 57.600  64.800 1.07165 61.0024304 61.0009438 0.8517477 
 74 58.400  63.200 1.04797 61.5204700 61.5189708 0.8589808 
 75 59.200  61.600 1.02409 62.0265161 62.0250046 0.8660465 
 76 60.000  60.000 1.00000 62.5204700 62.5189464 0.8729434 
 77 60.800  58.400 0.97572 63.0022354 63.0007001 0.8796700 
 78 61.600  56.800 0.95125 63.4717184 63.4701717 0.8862252 
 79 62.400  55.200 0.92659 63.9288275 63.9272696 0.8926076 
 80 63.200  53.600 0.90176 64.3734735 64.3719048 0.8988159 
 81 64.000  52.000 0.87674 64.8055698 64.8039905 0.9048491 
 82 64.800  50.400 0.85156 65.2250321 65.2234426 0.9107059 
 83 65.600  48.800 0.82621 65.6317787 65.6301793 0.9163851 
 84 66.400  47.200 0.80070 66.0257302 66.0241212 0.9218856 
 85 67.200  45.600 0.77503 66.4068098 66.4051915 0.9272064 
 86 68.000  44.000 0.74921 66.7749434 66.7733161 0.9323465 
 87 68.800  42.400 0.72325 67.1300590 67.1284231 0.9373048 
 88 69.600  40.800 0.69714 67.4720874 67.4704432 0.9420804 
 89 70.400  39.200 0.67090 67.8009621 67.7993098 0.9466723 
 90 71.200  37.600 0.64453 68.1166188 68.1149589 0.9510797 
 91 72.000  36.000 0.61803 68.4189961 68.4173288 0.9553016 
 92 72.800  34.400 0.59142 68.7080349 68.7063606 0.9593374 
 93 73.600  32.800 0.56468 68.9836790 68.9819979 0.9631860 
 94 74.400  31.200 0.53784 69.2458746 69.2441871 0.9668469 
 95 75.200  29.600 0.51089 69.4945705 69.4928770 0.9703194 
 96 76.000  28.000 0.48384 69.7297184 69.7280191 0.9736026 
 97 76.800  26.400 0.45670 69.9512723 69.9495676 0.9766961 
 98 77.600  24.800 0.42947 70.1591890 70.1574793 0.9795991 
 99 78.400  23.200 0.40216 70.3534281 70.3517136 0.9823112 
100 79.200  21.600 0.37476 70.5339517 70.5322328 0.9848318 
101 80.000  20.000 0.34730 70.7007245 70.6990015 0.9871603 
102 80.800  18.400 0.31976 70.8537140 70.8519874 0.9892964 
103 81.600  16.800 0.29217 70.9928905 70.9911605 0.9912397 
104 82.400  15.200 0.26451 71.1182268 71.1164937 0.9929897 
105 83.200  13.600 0.23681 71.2296985 71.2279626 0.9945461 
106 84.000  12.000 0.20906 71.3272837 71.3255455 0.9959087 
107 84.800  10.400 0.18127 71.4109636 71.4092234 0.9970771 
108 85.600   8.800 0.15344 71.4807218 71.4789799 0.9980511 
109 86.400   7.200 0.12558 71.5365447 71.5348014 0.9988305 
110 87.200   5.600 0.09770 71.5784213 71.5766770 0.9994152 
111 88.000   4.000 0.06980 71.6063437 71.6045987 0.9998050 
112 88.800   2.400 0.04188 71.6203062 71.6185608 1.0000000 
[ 71.618560834737543729429789502783231967988796348224, 1.0]  <=== 1.0 ratio indicates 
113 89.600   0.800 0.01396 71.6203062 71.6185608 1.0000000                exact match
[ 71.618560834737543729429789502783231967988796348224, 1.0]  to 50 significant digits
114 90.400  -0.800 0.01396 71.6342687 71.6325230 1.0001950 

[ 71.632523015076689001051700890480007430476705061781,

1.0001949519813915768975237841226157112052042164957]

[ 71.632523015076689001051700890480007430476705061781,

1.0001949519813915768975237841226157112052042164957]

>>

Edited on March 11, 2021, 10:01 am
  Posted by Charlie on 2021-03-11 10:00:29

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