Circumference 9811

Kristýna chose a certain odd natural number divisible by three. Jakub and David then examined triangles with a circumference in millimeters equal to the number selected by Kristýna and whose sides have lengths in millimeters expressed by different integers.
Jakub found a triangle in which the longest side had the longest possible length and wrote this value on the board. David found a triangle in which the shortest side has the longest possible length and wrote this value on the board. Kristýna correctly added both measurements on the board and got 1,681 mm.
Determine which number Kristýna chose.

Correct answer:

K =  2019

Step-by-step explanation:

K = 3k, k  N   K = a+b+c a < b < c  3k = a+b+c  max = (K1)/2 min=1  K = j1+j2+j3 j1 < j2 < j3  j1 = 1  j1=2 j2 = (K3)/2 j3 = (K1)/2  1 + (K3)/2+(K1)/2 = K  K = d1+d2+d3 d1 < d2 < d3 d1 = K/31 d2 = K/3+0 d3 = K/3+1  d1+j3 = 1681 K/31 + (K1)/2 = 1681  3k/31+(3k1)/2=1681  3 k/31+(3 k1)/2=1681  15k=10095  k=1510095=673  k=673  K=3 k=3 673=2019  j1=2 mm j2=(K3)/2=(20193)/2=1008 mm j3=(K1)/2=(20191)/2=1009 mm o1=j1+j2+j3=2+1008+1009=2019 mm  d1=K/31=2019/31=672 mm d2=K/3=2019/3=673 mm d3=K/3+1=2019/3+1=674 mm o2=d1+d2+d3=672+673+674=2019 mm  o1=o2 = K  x1=d1+j3=672+1009=1681  K=2019

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