Triangle circumference puzzle

Kristýna chose a certain odd natural number divisible by three. Jacob 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.

Final Answer:

K =  2019

Step-by-step explanation:

$$\begin{gathered} K=3k,k\in N \\ K=a+b+c \\ a<b<c \\ 3k=a+b+c \\ \max =(K-1)/2 \\ min=1 \\ K=j_1+j_2+j_3 \\ {j}_1<j_2<j_3 \\ {j}_1=1\vee j_1=2 \\ {j}_2=(K-3)/2 \\ {j}_3=(K-1)/2 \\ 1+(K-3)/2+(K-1)/2=K \\ K=d_1+d_2+d_3 \\ {d}_1<d_2<d_3 \\ {d}_1=K/3-1 \\ {d}_2=K/3+0 \\ {d}_3=K/3+1 \\ {d}_1+j_3=1681 \\ K/3-1+(K-1)/2=1681 \\ 3k/3-1+(3k-1)/2=1681 \\ 3\cdot k/3-1+(3\cdot k-1)/2=1681 \\ 15k=10095 \\ k=\frac{10095}{15}=673 \\ k=673 \\ K=3\cdot k=3\cdot 673=2019 \\ {j}_1=2\text{mm} \\ {j}_2=(K-3)/2=(2019-3)/2=1008\text{mm} \\ {j}_3=(K-1)/2=(2019-1)/2=1009\text{mm} \\ {o}_1=j_1+j_2+j_3=2+1008+1009=2019\text{mm} \\ {d}_1=K/3-1=2019/3-1=672\text{mm} \\ {d}_2=K/3=2019/3=673\text{mm} \\ {d}_3=K/3+1=2019/3+1=674\text{mm} \\ {o}_2=d_1+d_2+d_3=672+673+674=2019\text{mm} \\ {o}_1=o_2=K \\ {x}_1=d_1+j_3=672+1009=1681 \\ K=2019 \end{gathered}$$

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