11.5

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

1995th 7-digit number (II Soros Olympiad 1995-96 R1 11.5)

Source:

Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers

1

2

3

4

5

6

7

are used once each. Let's number these numbers in ascending order. What number will be the

1995th

number theoryDigits

polyhedron has exactly 6 right dihedral angles.

Source: II Soros Olympiad 1995-96 R2 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6

points are taken on the surface of the sphere, forming three pairs of diametrically opposite points on the sphere. Consider a convex polyhedron with vertices at these points. Prove that if this polyhedron has one right dihedral angle, then it has exactly

6

right dihedral angles.

geometry3D geometryanglespolyhedron

plane intersevting uinit cubes, filling space

Source: II Soros Olympiad 1995-96 R3 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

The space is filled in the usual way with unit cubes. (Each cube is adjacent to

6

others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of

1/19

1/9

1/7

from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?

combinatoricscombinatorial geometry