Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.
1. **Setup the Problem:**
   We are given an odd number of vectors, each of length 1, originating from a common point \( O \) and all situated in the same semi-plane determined by a straight line passing through \( O \). We need to prove that the sum of these vectors has a magnitude of at least 1.

2. **Choose Coordinates:**
   Without loss of generality, we can place the origin \( O \) at \((0,0)\) and assume that all vectors lie in the region \( y \geq 0 \). This means that all vectors have non-negative \( y \)-coordinates.

3. **Sum of Vectors:**
   Let the vectors be \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2n+1} \), where \( 2n+1 \) is an odd number. We need to show that the magnitude of their sum is at least 1.

4. **Sum of First \( 2n \) Vectors:**
   Consider the sum of the first \( 2n \) vectors:
   \[
   \mathbf{w} = \mathbf{v}_1 + \mathbf{v}_2 + \cdots + \mathbf{v}_{2n}
   \]
   Since each vector has a length of 1 and lies in the semi-plane \( y \geq 0 \), the vector \( \mathbf{w} \) will also lie in the same semi-plane.

5. **Adding the Last Vector:**
   Let \( \mathbf{v}_{2n+1} \) be the last vector. We need to minimize \( |\mathbf{w} + \mathbf{v}_{2n+1}| \), where \( |\mathbf{v}_{2n+1}| = 1 \) and \( \mathbf{v}_{2n+1} \) has a positive \( y \)-coordinate.

6. **Locus of \( \mathbf{w} + \mathbf{v}_{2n+1} \):**
   The possible locus of \( \mathbf{w} + \mathbf{v}_{2n+1} \) is a semicircle \( \Gamma \) of unit radius centered at \( \mathbf{w} \) and located above the origin. The closest point of \( \Gamma \) to the origin is one of the endpoints of the semicircle.

7. **Minimizing the Distance:**
   To minimize the distance \( |\mathbf{w} + \mathbf{v}_{2n+1}| \), we consider the endpoints of the semicircle. Since \( \mathbf{v}_{2n+1} \) has a positive \( y \)-coordinate, the minimum distance from the origin to the semicircle is 1.

8. **Conclusion:**
   Therefore, the sum of the \( 2n+1 \) vectors, each of length 1 and all situated in the same semi-plane, has a magnitude of at least 1.

\[
\boxed{1}
\]