VTMO 2026 - Grade 10 - Semifinal - SET 1 - Self-study Question 5

Positive real numbers $a,b,c$ satisfy $a^2+b^2+c^2=3$. Find the smallest real number $k$ such that whenever $\min\{a+b,b+c,c+a\}>k$, the numbers $a,b,c$ are the side lengths of a triangle.