11.4

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

infinite queens on an infinite chessboard (I Soros Olympiad 1994-95 R1 11.4)

Source:

Given a chessboard that is infinite in all directions. Is it possible to place an infinite number of queens on it so that on each horizontally, on each vertical and on each diagonal of both directions (i.e. on a set of cells located at an angle of

45^o

135^o

to the horizontal) was exactly one queen?

combinatoricsChessboard

all points at distance <=1 from a square (I Soros Olympiad 1994-99 Round 2 11.4)

Source:

The wire is bent in the form of a square with side

2

. Find the volume of the body consisting of all points in space located at a distance not exceeding

1

from at least one point of the wire.

geometry3D geometry

A_1X / A_1O +B_1X/B_1O +C_1X/C_1O +D_1X/D_1O =4, insphere of tetrahedron

Source: I Soros Olympiad 1994-95 Ukraine R2 11.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

ABCD

is given, in which each pair of adjacent edges are equal segments. Let

O

be the center of the sphere inscribed in this tetrahedron .

X

is an arbitrary point inside the tetrahedron,

X \ne O

OX

intersects the planes of the faces of the tetrahedron at the points marked by

A_1

B_1

C_1

D_1

\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4

geometry3D geometryinspheretetrahedron