Does there exist a Riemannian metric on the $n$-sphere ($n > 2$) such that at each point some (but not every) sectional curvature is negative? 


For $n=2$ it is easily seen that such a metric cannot exist.



Riemannian metric on the sphere with at least one negative sectional curvature at every point
Joachim Lohkamp has shown ([Annals of Mathematics, 1994](https://www.jstor.org/stable/2118620)) that any smooth manifold of dimension greater than two admits a metric of negative Ricci curvature. So it seems that the answer to your question is yes. 


