Let $f \colon \mathbb{CP}^n \to \mathbb{CP}^n$ be a continuous function which induces a non-zero map $f_*$ on every Homology group $H_{2k} (\mathbb{CP}^n)$. Show that $f$ is surjective.
As we all know $H_{2k} (\mathbb{CP}^n) \cong \mathbb{Z}$, if $n \geq k$.
Thus if we look at the $2k$-th homology, $f_* : \mathbb{Z} \xrightarrow{} \mathbb{Z}$ must be a multiplication by a scalar. i.e. $f_*(x) = c_k x$ (for $x$ the generator of $H_{2k} (\mathbb{CP}^n)$). By the assumption $c_k \neq 0$ for all $k \leq n$.
What else can we say?