  Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such
that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of
independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove
that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an
interval of the form [x, \infty) or just a single point. The inequality is
exact and the optimal values of c and k are given explicitly. It improves
Kwapie\'n's inequality in the case of the Rademacher series. We also provide a
new and very short proof of the Littlewood-Offord problem without using
Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.
