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In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary-value problem $$displaylines{ u''+f(t,u)=0,quad 0 less than t lessthan 1, cr u'(0)=0,quad u(1)=alpha u'(eta). }$$ where $eta in (0,1)$, $alpha in mathbb{R}$, $fin C([0,1]imes mathbb{R},mathbb{R})$. Under certain growth conditions on the nonlinearity $f$ and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of nontrivial solution are obtained. We illustrate the results obtained with some examples.