I want to propose this problem.
Suppose that $C:F(X,Y,Z)=0,\;C':G(X,Y,Z)=0$ are two plane curves defined over a number field $K$. Suppose that they have no common component and that all the intersection points are defined over $K$. Let $\mathfrak{p}$ a prime of the ring of integers $\mathcal{O}_K$. Let $\kappa(\mathfrak{p})$ the residue field. Assume that $F,G\in \mathcal{O}_\mathfrak{p}[X,Y,Z]$ have coprime coefficients, denote by $\bar{F}$ and $\bar{G}$ the reduced polynomials in $\kappa(\mathfrak{p})[X,Y,Z]$.
Let $\bar{C}$ and $\bar{C'}$ the curves in $\mathbb{P}^n(\overline\kappa(\mathfrak{p}))$ respectively defined by the equations $\bar{F}(X,Y,Z)=0$ and $\bar{G}(X,Y,Z)=0$. Suppose that $\bar{C}$ and $\bar{C'}$ have no common component.
CLAIM: All the intersection points of $\bar{C}\cap\bar{C'}$ can be lifted to intersection points of $C\cap C'$.
My idea: The number of intersection points (counting multiplicities) is $\deg(\bar{C})\deg(\bar{C'})=\deg(C)\deg(C')$ by Bezout theorem. So, if we are able to prove that the reduction map preserves the intersection numbers, we have finished. For "preserves the intersection numbers" i mean that if $P_1,..,P_t$ are all the points of $C\cap C'$ sent to $\bar{P}\in \bar{C}\cap\bar{C'}$ by the reduction map, then $$I(\bar{P},\bar{C}\cap\bar{C'})=\sum_{i=1}^t I(P_i,C\cap C') %$$ This fact is somewhat intuitive, but i'm not able to formalize.
Thanks to every one who will help me.