2. In the triangular pyramid $SABC$, points $\boldsymbol{D}$, $\boldsymbol{E}$, and $\boldsymbol{F}$ are chosen on the edges $SA$, $SB$, and $SC$ respectively, such that the area of triangle $\boldsymbol{ABE}$ is $1/3$ of the area of triangle $\boldsymbol{AES}$, the area of triangle $\boldsymbol{BCF}$ is $1/4$ of the area of triangle $BFS$, and the area of triangle $ACD$ is $1/5$ of the area of triangle DCS. Find the ratio of the volumes of pyramids SDEF and $SABC$. (14 points).
# Solution:

Let there be a pyramid $SABC$ and a point $D$ on side $SA$, such that $SD:SA = d$, on side

![](https://cdn.mathpix.com/cropped/2024_05_06_920a3f7017259ebe2bb7g-2.jpg?height=69&width=1647&top_left_y=705&top_left_x=273)
the volume ratio of $SDEF$ to $SABC$ is $\boldsymbol{def}$. Indeed, let's "lay down" the pyramid so that the base is the triangle $\boldsymbol{SAB}$. The ratio of the area of triangle $SDE$ to the area of $SAB$ is $de$. The heights of the pyramids $SDEF$ and $SABC$, dropped from points $\boldsymbol{F}$ and $\boldsymbol{C}$ respectively, are in the ratio $\boldsymbol{f}$. Therefore, the ratio of the volumes of the pyramids is the product of the ratio of the areas of the bases and the ratio of the heights, i.e., $def$. Thus, (3/4)·(4/5)·(5/6) = 0.5. Answer: 0.5.

| points | Criteria for evaluating the performance of task No. 2 |
| :---: | :---: |
| 14 | A correct and justified sequence of all steps of the solution is provided. All transformations and calculations are performed correctly. The correct answer is obtained. |
| 10 | Gaps are allowed in the reasoning about the heights of the pyramids or the further course of the solution, resulting in an incorrect answer. |
| 5 | ![](https://cdn.mathpix.com/cropped/2024_05_06_920a3f7017259ebe2bb7g-2.jpg?height=340&width=1470&top_left_y=1722&top_left_x=455) |
| 0 | All cases of the solution that do not correspond to the criteria for awarding 5 or 10 points. |