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 = 3 k, k \in N K = a + b + c a < b < c 3 k = a + b + c max = (K - 1) / 2 m i n = 1 K = j_{1} + j_{2} + j_{3} j_{1} < j_{2} < j_{3} j_{1} = 1 \lor 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 3 k / 3 - 1 + (3 k - 1) / 2 = 1681 3 \cdot k / 3 - 1 + (3 \cdot k - 1) / 2 = 1681 15 k = 10095 k = \frac{1 0 0 9 5}{1 5} = 673 k = 673 K = 3 \cdot k = 3 \cdot 673 = 2019 j_{1} = 2 mm j_{2} = (K - 3) / 2 = (2019 - 3) / 2 = 1008 mm j_{3} = (K - 1) / 2 = (2019 - 1) / 2 = 1009 mm o_{1} = j_{1} + j_{2} + j_{3} = 2 + 1008 + 1009 = 2019 mm d_{1} = K / 3 - 1 = 2019 / 3 - 1 = 672 mm d_{2} = K / 3 = 2019 / 3 = 673 mm d_{3} = K / 3 + 1 = 2019 / 3 + 1 = 674 mm o_{2} = d_{1} + d_{2} + d_{3} = 672 + 673 + 674 = 2019 mm o_{1} = o_{2} = K x_{1} = d_{1} + j_{3} = 672 + 1009 = 1681 K = 2019

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