  We investigate the existence of positive solutions to the nonlinear
second-order three-point integral boundary value problem \begin{equation*}
\label{eq-1} \begin{gathered} {u^{\prime \prime}}(t)+f(t, u(t))=0,\ 0<t<T, \\
u(0)={\beta}u(\eta),\ u(T)={\alpha}\int_{0}^{\eta}u(s)ds, \end{gathered}
\end{equation*} where $0<{\eta}<T$, $0<{\alpha}< \frac{2T}{{\eta}^{2}}$,
$0<{\beta}<\frac{2T-\alpha\eta^{2}}{\alpha\eta^{2}-2\eta+2T}$ are given
constants. We establish the existence of at least three positive solutions by
using the Leggett-Williams fixed-point theorem.
