Folklore

What is the smallest value that the perimeter of a scalene triangle with integer side lengths can take?

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Let $a, b$ and $c$ be the integer lengths of the sides of a triangle and $a>b>c$. According to the triangle inequality, $c>a-b$. Since $a$ and $b$ are different natural numbers, $c \geq 2$, thus $b \geq 3$ and $a \geq 4$. Therefore, $a+b+c \geq 9$. Equality is achieved for a triangle with sides 2, 3, and 4.

## Otвет

9.