IT-CESENATICO 1985

Let $x_1 = 97$ , and for $n > 1$ let $x_n = \frac{n}{x_{n - 1}}$ . Calculate the product $x_1 x_2 \dotsm x_8$ .

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$ . When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$ . What is the length (in cm) of the hypotenuse of the triangle?

A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.

$1/n$

As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$ .

$A$ $B$ $C$ 84353040