The following is Exercise 6.7.22 from Riemannin Geometry by P. Petersen.
Use an analog of theorem 6.2.3 to show that any closed manifold of constant curvature $=1$ must either be the standard sphere or have diameter $\leq\pi/2$. Generalize this to show that any closed manifold with $\operatorname{sec}\geq1$ is either simply connected or has diameter $\leq\pi/2$.
Theorem 6.2.3 is Cartan's theorem asserting that an isometry of a simply connected complete manifold of nonpositive curvature which has finite order must have a fixed point. The proof uses the well-known center of mass construction and exploits the convexity of distance functions in nonpositive curvature.
How does this generalize to positive curvature? I have no idea what to do. Any hints will be appreciated!