5. One base of a regular $n$-sided prism ( $n \geqslant 3$ ) has $n$ common points with a sphere of radius 3; the other base has one common point with this sphere. What values can the volume of the prism take?
Solution. If the base of the prism has $n$ common points with the sphere, then this base is inscribed in the corresponding circle. Then the second base touches the sphere at the center of the base. We will describe a cylinder around the prism (its volume is greater than the volume of the prism) and look for possible values of the cylinder's volume.

Let the radius of the sphere be $R$, and the height of the cylinder be $R+x$, where $x \in(-R ; R)$. Then the radius of the cylinder is $\sqrt{R^{2}-x^{2}}$, and its volume is $V=\pi\left(R^{2}-x^{2}\right)(R+x)$. We will investigate this function for $x \in(-R ; R)$. Since $V^{\prime}(x)=\pi\left(-2 x R-2 x^{2}+R^{2}-x^{2}\right)$ $=-\pi\left(3 x^{2}+2 R x-R^{2}\right)=-3 \pi(x+R)\left(x-\frac{R}{3}\right)$, the maximum of $V(x)$ is achieved at $x=\frac{R}{3}$ and is equal to $\frac{32}{27} \pi R^{3}$, the minimum tends to zero as $x \rightarrow-R$ (when the height of the cylinder tends to zero) and as $x \rightarrow R$ (when the radius of the cylinder tends to zero). Therefore, for the cylinder, the desired ratio takes values from the interval $\left(0 ; \frac{32}{27} \pi R^{3}\right]$. Therefore, for the pyramid, we have the interval $\left(0 ; \frac{32}{27} \pi R^{3}\right)$ (the ratio approaches the value $\frac{32}{27} \pi R^{3}$ arbitrarily closely as $\left.n \rightarrow \infty\right)$.

Answers in different versions: $(0 ; 32 \pi) ;\left(0 ; \frac{32 \pi}{9}\right) ;(0 ; 256 \pi) ;\left(0 ; \frac{64 \pi}{9}\right)$.

Those who obtained the correct answer not for an arbitrary, but for a given value of $n$, the task was also counted, as in the limit, the same answer is obtained.

## 2013/2014 Academic Year CRITERIA FOR DETERMINING WINNERS AND PRIZE WINNERS ${ }^{1}$

## of the schoolchildren's olympiad "CONQUER THE SPARROW MOUNTAINS!" IN MATHEMATICS

ELIMINATION STAGE

WINNER:

From 95 points inclusive and higher.

PRIZE WINNER:

From

91 points to

94

points inclusive.

FINAL STAGE

WINNER (Diploma I degree):

From 90 points inclusive and higher.

PRIZE WINNER (Diploma II degree):

From 75 points to 89 points inclusive.

PRIZE WINNER (Diploma III degree):

from 60 points to 74 points inclusive.[^0]


[^0]:    ${ }^{1}$ Approved at the meeting of the jury of the schoolchildren's olympiad "Conquer the Sparrow Mountains!" in mathematics