An Aztec pyramid is a regular quadrilateral frustum with a base edge of $81 \mathrm{~m}$, a side edge of $65 \mathrm{~m}$, and a top edge of $16 \mathrm{~m}$. A ramp is planned for tourists, starting at one corner of the base, passing along all four sides, and ending at the top corner of the top face, rising evenly throughout. At which points must the ramp touch the side edges?
It is easy to see that every side of the frustum is a trapezoid with base angles of $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_89e6d581346c5d242fa1g-1.jpg?height=459&width=615&top_left_y=215&top_left_x=300)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_05_02_89e6d581346c5d242fa1g-1.jpg?height=623&width=680&top_left_y=713&top_left_x=1140)

Let's draw the segments of the ascent one after another in the trapezoid $ABCD$, starting from $A$. The angle of ascent is $\alpha$, the first segment intersects the side $BC$ at $P$, the second segment intersects the side $AD$ at $Q$, the third segment intersects the side $BC$ at $R$, and the final point is $D$ (Figure 2). Draw lines parallel to the base of the trapezoid through points $P$, $Q$, and $R$, intersecting the opposite side at points $P_{1}$, $Q_{1}$, and $R_{1}$, respectively. Denote the segments $PP_{1}$, $QQ_{1}$, and $RR_{1}$ by $x$, $y$, and $z$, respectively.

The triangles $ABP$, $PP_{1}Q$, $Q_{1}QR$, and $R_{1}RD$ are similar (their angles are equal), so we can write the following ratios for their sides:

$$
81: x = x: y = y: z = z: 16
$$

From this,

$$
x^{2} = 81y, \quad y^{2} = xz, \quad z^{2} = 16y
$$

which gives

$$
\frac{x^{2}}{z^{2}} = \frac{81}{16}
$$

and since $x$, $y$, and $z$ are positive,

$$
\frac{x}{z} = \frac{9}{4}, \quad y^{2} = \frac{9}{4}z^{2}, \quad y = \frac{3}{2}z
$$

From $z^{2} = 16 \cdot \frac{3}{2}z$,

$$
z = 24, \quad y = 36, \quad x = 54
$$

With these segments known, we can determine the distances $BP$, $BQ_{1}$, and $BR$.

$BP = AB - PP_{1} = 27$, and similarly $PQ_{1} = 18$, $Q_{1}R = 12$, and $RD = 8$. Therefore, the points of contact between the ascent and the side edges are at distances of $27$, $45$, and $57 \, \text{m}$ from the lower end of the corresponding side edge.