Is there a topological space $X$ that is not contractible and for which every mapping $S^n \rightarrow X$ for $n \geq 1$ is homotopic to a constant mapping?
It seems to me that such an $X$ does not exist because $S^n$ is not contractible and, intuitively speaking, to "homotope" any such map to a constant map, you would need one of the spaces to be contractible. And since $S^n$ is not contractible, $X$ must be.
However, I fail to prove this. I was trying to construct a homotopy between $id_X$ and a constant map from homotopies between $S^n$ and $X$. But I totally fail to do so. Can someone help me?