Summary: This paper is concerned with the nonlinear third-order boundary value problem \[ u_i'''(t) + f_i(t, u_1(t), \dots, u_n(t), u_1'(t), \dots, u_n'(t)) = 0, \quad 0 < t < 1 \] with three-point and integral conditions \[ \begin{cases} u_i(0)= \int_0^1 h_{1,i}(s,u_1(s), \dots, u_n(s))ds, \\ u_i'(0)=0, \\ \alpha_iu_i'(1) = \beta_iu_i(\eta_i) + \int_0^1 h_{2,i}(s,u_1(s), \dots, u_n(s))ds, \end{cases} \] where \(i \in \{1, \dots, n\}\), \(\alpha_i > 0\), \(\beta_i > 0\), \(1 > \eta_i \geq 0\), \(2 \alpha_i > \beta_i \eta_i^2\), \(f_i : [0, 1] \times \mathbb R^n \times \mathbb R^n \to \mathbb R\), for all \(k \in \{1, 2\}\), \(h_{k,i} : [0, 1] \times \mathbb R^n \to \mathbb R\) are continuous functions. By using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solution of this problem. We also prove nonexistence of positive solution and we give some examples to illustrate our results.