In a triangle, two sides $a$ and $b$ are known. What should the third side be so that the largest angle of the triangle has the smallest possible value?

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Answer: the third side should be equal to the greatest of the sides $a$ and $b$. Let for definiteness $a \geq b$. Consider a semicircle of radius $a$. Let $A$ be the center of this semicircle, $C$ be a point on the diameter such that $A C=b$. The vertex $B$ is located on this semicircle. The greatest side of the triangle is $a$ or $c$, so the greatest angle is $A$ or $C$. Choose the vertex $B$ so that it lies on the perpendicular bisector of the segment $A C$ (this is equivalent to $a=c$). If the vertex $B$ moves along the semicircle from this position, then either angle $A$ or angle $C$ increases (depending on which direction it moves).