Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.
1. **Base Case:**
   - Consider the case when \( d = 1 \). In one-dimensional space, any two non-null vectors (rays) will either point in the same direction or in opposite directions. If they point in opposite directions, they form an angle of \( 180^\circ \), which is not acute. Therefore, the largest number of vectors in \(\mathbb{R}^1\) with pairwise non-positive dot products is 2.

2. **Inductive Step:**
   - Assume that in \( \mathbb{R}^d \), the largest number of non-null vectors with pairwise non-positive dot products is \( 2d \).
   - Consider \( \mathbb{R}^{d+1} \) and let \( u \) be one of the vectors. Denote by \( U = \langle u \rangle^{\perp} \) the orthogonal complement of \( u \), which has dimension \( d \).

3. **Decomposition of Vectors:**
   - Any vector \( v \) in \( \mathbb{R}^{d+1} \) can be uniquely written as \( v = v_{\perp} + v_{\parallel} \), where \( v_{\parallel} = \frac{\langle u, v \rangle}{||u||^2} u \) and \( v_{\perp} \perp u \), i.e., \( v_{\perp} \in U \).
   - The coefficients \( \frac{\langle u, v \rangle}{||u||^2} \) are non-positive under our conditions. Notice that \( v_{\perp} = 0 \) can occur only once, for some \( u' = v \) (trivially so, since then \( u' = u'_{\parallel} = \lambda u \) for some negative real \( \lambda \)).

4. **Orthogonal Components:**
   - Consider the orthogonal components \( v_{\perp} \in U \). We have:
     \[
     \langle v, w \rangle = \langle v_{\perp} + v_{\parallel}, w_{\perp} + w_{\parallel} \rangle = \langle v_{\perp}, w_{\perp} \rangle + \langle v_{\perp}, w_{\parallel} \rangle + \langle v_{\parallel}, w_{\perp} \rangle + \langle v_{\parallel}, w_{\parallel} \rangle
     \]
     \[
     = \langle v_{\perp}, w_{\perp} \rangle + \frac{\langle u, v \rangle \langle u, w \rangle}{||u||^2}
     \]
   - Therefore,
     \[
     \langle v_{\perp}, w_{\perp} \rangle = \langle v, w \rangle - \frac{\langle u, v \rangle \langle u, w \rangle}{||u||^2} \leq 0
     \]
     since \( \langle v, w \rangle \leq 0 \) and \( \langle u, v \rangle \langle u, w \rangle \geq 0 \).

5. **Induction Hypothesis:**
   - By the induction hypothesis, the number of non-null vectors in \( U \) is at most \( 2d \). Including \( u \) and possibly \( u' \), there are at most \( 2(d + 1) \) non-null vectors in \( \mathbb{R}^{d+1} \).

6. **Model Construction:**
   - To achieve the bound, we must have the pair \( u \) and \( u' \), with all other vectors orthogonal to them. By inductive reasoning, the only possible model is made by some \( d \) pairwise orthogonal non-null vectors, and some other \( d \) obtained from them by multiplication with arbitrary negative scalars.

7. **Conclusion:**
   - The largest number of vectors in \( \mathbb{R}^d \) with pairwise non-positive dot products is \( 2d \). Therefore, in \( \mathbb{R}^3 \), the largest number of such vectors is \( 2 \times 3 = 6 \).

8. **Acute Angle Condition:**
   - To ensure that among any \( n \) rays in space sharing a common origin there exist two which form an acute angle, we need \( n > 6 \). Thus, the least \( n \) is \( 7 \).

The final answer is \( \boxed{7} \).