4. In a circle with a radius of 1, there is an inscribed polygon. If all its sides are greater than 1 but less than $\sqrt{2}$, then the number of sides of this polygon must be
(A) 7 ;
(B) 6 ;
(C) 5 ;
(D) 4 .
4. (C)

Solution If the number of sides of the polygon that satisfies the condition is less than or equal to 6, then the central angle opposite at least one side is no greater than $60^{\circ}$. By the cosine rule, the length of this side must be no greater than 1; similarly, if there exists a quadrilateral that satisfies the condition, then it must have at least one side whose length is no less than $\sqrt{2}$.