5. All edges of a regular quadrilateral pyramid $S A B C D$ with base $A B C D$ have equal length. Plane $\alpha$ is perpendicular to line $S A$, and plane $\beta$ is parallel to line $C D$. Determine the smallest possible angle between planes $\alpha$ and $\beta$. (7 points) Answer: $\frac{\pi}{6}$.
Solution: A similar solution to a similar problem is present in variant 1 under the same number; however, for this problem, let's note the following: Triangle $A S C$ is a right triangle. Let plane $\alpha$ pass through point $S$, then line $S C \in \alpha$. If we draw plane $\beta$ through line $C D$ and assume that all edges are equal to 1, then the length of the perpendicular dropped from $D$ to $\alpha$ is $\frac{1}{2} A S = \frac{1}{2}$. $C D$ is the segment connecting $D$ and the edge of the dihedral angle, $C D = 1$. If $\varphi$ is the dihedral angle between planes $\alpha$ and $\beta$, then $\sin \varphi \geq \frac{1}{2}$.

Evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| **7** | Complete and justified solution. |
| **6** | Justified solution with minor flaws. |
| **5 - 6** | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct with minor corrections or additions. |
| **4** | The problem is more solved than not, for example, one of the two (more complex) significant cases is correctly considered. |
| **2 - 3** | The problem is not solved, but formulas, drawings, considerations, or auxiliary statements relevant to the solution of the problem are provided. |
| **1** | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or in the case of an incorrect solution. |
| **0** | The solution is absent, or the solution does not meet any of the criteria listed above. |

10th grade. Problem 5.
![](https://cdn.mathpix.com/cropped/2024_05_06_8cf98bbe6aaef78ad6dcg-6.jpg?height=478&width=1400&top_left_y=548&top_left_x=435)

Fig. 5

Fig. 4

Fig. 6

11th grade. Problem 5.

![](https://cdn.mathpix.com/cropped/2024_05_06_8cf98bbe6aaef78ad6dcg-6.jpg?height=377&width=505&top_left_y=1533&top_left_x=524)

Fig. 7

![](https://cdn.mathpix.com/cropped/2024_05_06_8cf98bbe6aaef78ad6dcg-6.jpg?height=480&width=422&top_left_y=1459&top_left_x=1271)

Fig. 8