新浪博客
HuangZhen
第58届IMO
中国国家队左起：姚一隽老师，周青老师，江元旸，任秋宇，何天成，李大潜院士，周行健，张騄，吴金泽，熊斌老师。
1.
I think the proof is not correct, as we can get 9 visible edges by taking a vantage point above the center of one of the faces like shown here:
Figure 1
The Octahedron has 12 sides, 8 faces and 6 vertices. Count them!
Each of the octahedron's 8 faces is an equilateral triangle, just like the tetrahedron, but the tetrahedron only has 4 faces.
Notice how the octahedron can be considered to be formed from 3 orthogonal squares:
The square BCDE, the square ABFD, and the square ACFE, all 3 of which are planes and all 3 of which are perpendicular to each other.
The octahedron's face angles are all equal because they are all 60°, however, the internal angles of the squares are all 90°. I guess it depends on which way you want to look at it! That's spatial geometry: the position of the observer relative to an object can yield quite different perspectives.
Let's get the volume of the octahedron. All of the octahedron's vertices touch upon the surface of the same sphere that encloses the tetrahedron, so lets put a centroid at O. We will also divide up the face CDF and place a midpoint G in the middle of that face. We place a midpoint H on CD, one of the sides of the octahedron. The side or edge of the octahedron will be hereinafter referred to as ‘os.’
Figure 2
First lets find the volume:
The octahedron consists of 2 pyramids, face-bonded. One pyramid is at A-BCDE, the other at F-BCDE.
The base of each pyramid is the square BCDE. The area of the base is then just os * os = os².
The height of the pyramids are
But OF = OA = OC, since the octahedron is composed of 3 squares. FC is just a side of the octahedron.
Therefore we can write:
