Let $n\\geq 1$ be an integer, and $\\mathcal L \\subset \\mathbb{R}^n$ be a compact smooth affine real hypersurface, not necessarily connected. We prove that there exist $c>0$ and $d0\\geq 1$ such that for any $d\\geq d_0$, any smooth complex projective hypersurface $Z$ in $\\mathbb{C} P^n$ of degree $d$ contains at least $c\\dim H{\*}(Z, \\mathbb{R})$ disjoint Lagrangian submanifolds diffeomorphic to $\\mathcal L$, where $Z$ is equipped with the restriction of the Fubini–Study symplectic form (Theorem 1.1). If moreover all connected components of $\\mathcal L$ have non-vanishing Euler characteristic, which implies that $n$ is odd, the latter Lagrangian submanifolds form an independent family in $H\_{n-1}(Z, \\mathbb{R})$ (Corollary 1.2). These deterministic results are consequences of a more precise probabilistic theorem (Theorem 1.23) inspired by a 2014 result by J.-Y. Welschinger and the author on random real algebraic geometry, together with quantitative Moser-type constructions (Theorem 3.4). For $n=2$, the method provides a uniform positive lower bound for the probability that a projective complex curve in $\\mathbb{C} P^2$ of given degree equipped with the restriction of the ambient metric has a systole of small size (Theorem 1.6), which is an analog of a similar bound for hyperbolic curves given by M. Mirzakhani (2013). In higher dimensions, we provide a similar result for the $(n-1)$-systole introduced by M. Berger (1972) (Corollary 1.14). Our results hold in the more general setting of vanishing loci of holomorphic sections of vector bundles of rank between 1 and $n$ tensored by a large power of an ample line bundle over a projective complex $n$-manifold (Theorem 1.20).