The paper is devoted to the study of oscillations of solutions of linear and nonlinear differential inequalities caused by delayed arguments. The results obtained here are not valid for corresponding ordinary differential inequalities without delayed arguments. For example, all solutions of the differential inequality \[ x^{(2n)}(t)sgn x(t)+p(t)\prod^{m}_{i=1}| x(g_ i(t))|^{\alpha_ i}\leq 0 \] are oscillatory, if \[ \sum^{m}_{j=1}\alpha_ j\liminf_{t\to \infty}\int^{t}_{g_ j(t)}p(s)\prod^{m}_{i=1}[g_ i(s)]^{(2n- 1)\alpha_ i}ds>\frac{(2n-1)!}{e}, \] where \(\alpha_ i\) are nonnegative numbers with \(\alpha_ 1+...+\alpha_ m=1\), the functions \(p,g_ i: [0,\infty)\to [0,\infty)\) are continuous, \(g_ i(t)\leq t\) and \(\lim_{t\to \infty}g_ i(t)=\infty\) \((i=1,...,m)\).