[ $\underline{\text { Induction in Geometry }} \quad]$
$[$ Integer Lattices (other) ]

Authors: Burenev A.I. Kanel-Belov A.Ya.

On a meadow in the shape of a square, there is a circular pit. A grasshopper is jumping around the meadow. Before each jump, it chooses a vertex and jumps in the direction of this vertex. The length of the jump is half the distance to this vertex.

Can the grasshopper land in the pit?
It is sufficient to prove the following statement. Let each side of the square have a length of 1 and be divided into $2^{\text {n }}$ equal parts ( $\mathrm{n} \geq 0$ ), and through the division points, draw lines parallel to the sides. Then the grasshopper can reach any of the $4^{\mathrm{n}}$ resulting cells.

For $\mathrm{n}=0$, the fact is trivial. We will perform the inductive transition from $\mathrm{n}$ to n+1. Consider one of the cells of size $4^{-}$

${ }^{n-1}$. Choose the closest vertex of the original square and perform a homothety with the center at this vertex and a coefficient of 2. Then the selected cell will transition to one of the cells of size $4^{- \text {-n }}$. By the induction hypothesis, the grasshopper can reach it. If it jumps now to half the distance to the indicated vertex, it will land in the required cell.

![](https://cdn.mathpix.com/cropped/2024_05_06_f4319cad953e6ec80578g-03.jpg?height=524&width=458&top_left_y=2099&top_left_x=800)

On the plane, a square with side $a$ is given. Find the volume of the body consisting of all points in space whose distance from the part of the plane bounded by the square is no more than $a$.

## Solution

Consider the cross-section of the body by a plane coinciding with the plane of the square (see figure). Here $A_{1} B_{1}\|A B\| D C|| D_{1} C_{1}$, $A_{2} D_{2} \| A D|| B C|| B_{2} C_{2}$. The given body will consist of a parallelepiped with a base equal to the given square, height $2 a$, four half-cylinders with a base radius of $a$ and height $a$, and four quarters of a sphere with radius $a$. Therefore, the volume of the body is $2 a \cdot a^{2}+4\left(a \cdot \pi a^{2} / 2\right)+4\left(1 / 4 \cdot 4 / 3 \pi a^{3}\right)=2 / 3 a^{3}(5 \pi+3)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f4319cad953e6ec80578g-04.jpg?height=660&width=669&top_left_y=430&top_left_x=703)

## Answer

$2 / 3 a^{3}(5 \pi+3)$.