Let $(M,g)$ Riemannian manifold of dimension $\geq3$ with sectional curvatures $\leq0$ and let $p\in M$ such that some sectional curvatures are $<0$ at $p$. Can we find an arbitrarily small geodesic triangle $pqr$, with angle $\frac{\pi}{2}$ at $p$, such that the sum of its angles is $<\pi$?
Motivation: This MO question.