8.4. The volume of a regular pyramid is $8 \pi$. Can its lateral edge be equal to 3.96?
Answer: No.

LOMONOSOV - 2019. MATHEMATICS. Grading Criteria (10-11 grades)

| Problem № 1 = $\mathbf{1 5}$ points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| Correct equation, but arithmetic errors | $\mp$ | 5 |
| Note. Graphical solution is considered a solution. |  |  |


| Problem № 2 = 15 points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| Conceptually correct solution. (irrationality is correctly resolved, a and b are found) with full justifications, but the answer is incorrect due to one computational error | $\pm$ |  |
| Irrationality is correctly resolved, a and b are found, but there is no justification | ‡ | 10 |
| Note. | 干 | 5 |


| Problem № 3 = 15 points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| Arithmetic error at the end of the solution | $\pm$ | 10 |
| Negative root of the exponential equation is not discarded |  | 0 |


| Problem № 4 = 15 points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| One arithmetic error in a completely correct solution | $\pm$ | 10 |
| Arithmetic errors in a correct solution approach or the correct ratio of areas (segments) is found when the task is to find the ratio of segments (areas) | $\mp$ | 5 |
| Note. A solution using Menelaus' theorem (proof is not required) is allowed |  |  |


| Problem № 5 = 15 points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| The answer "yes" is correctly justified. The minimum value of n is found incorrectly with a correct solution approach. | $\pm$ | 10 |
| Only a one-sided estimate for f(n) is obtained | $\mp$ | 5 |
| Problem № 6 = 15 points | Pluses-minuses | Points |
| Monotonicity on half-intervals is proven, the problem is reduced to a system of two inequalities. The case where the values of trigonometric functions fall into different half-intervals is not considered | $\pm$ | 10 |
| Error in solving trigonometric inequalities with justified monotonicity. | $\mp$ | 5 |


| Problem № $\mathbf{7}=\mathbf{1 5}$ points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| Correct factorization, it is proven that each of the two equations in the equivalent system has a unique solution, and it is concluded that these solutions must coincide. | $\pm$ | 10 |
| Correct factorization. | $\mp$ | 5 |
| Note. "Proven" means mentioning the monotonic increase of the corresponding functions and indicating their range of values, or using convexity. If there is a correct answer confirmed by the graph of the logarithmic function, excluding tangency and the absence of solutions, then 15 points should be given |  |  |


| Problem № 8= 15 points | Pluses-minuses | Points |
| :--- | :---: | :---: |
| Reduced the problem to a cone, obtained the function for the volume of the cone, an error was made in the investigation of the function, possibly leading to an incorrect answer. Or without a cone. The extreme value of R (or the angle) is found, there is a formula for the maximum volume of the pyramid for each n and its exact upper bound is indicated | $\pm$ |  |
| Reduced the problem to a cone, obtained the function for the volume of the cone, but there is no or unproven required comparison | ‡ | 10 |