MathNet #0hsp
MathNetHard
From module Sphere Geometry
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Understand the scale
Scale
0.5: Easiest math competition problems, often solvable even for early middle school students without experience.
1: Problems strictly for beginners, often at a moderate middle school level.
1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests.
2: For motivated beginners, harder questions from the previous categories.
2.5: More advanced beginner problems, hardest questions from previous categories.
3: Early intermediate problems that require more creative thinking.
3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge.
4: Intermediate-level problems.
4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions.
5: More difficult AIME problems or simple proof-based Olympiad-style problems.
6: High-level AIME-styled questions or introductory Olympiad-level questions.
7: Tougher Olympiad-level questions, may require more technical knowledge.
8: High-level Olympiad-level questions.
9: Expert Olympiad-level questions.
9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve.
10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
Volume3D ShapesTrigonometryTriangle trigonometry
Problem
Problem:
Let be a dodecahedron which can be inscribed in a sphere with radius . Let be an icosahedron which can also be inscribed in a sphere of radius . Which has the greater volume, and why?
Note: A regular polyhedron is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with 12 faces which are regular pentagons and a regular icosahedron is a polyhedron with 20 faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere.
The illustration below shows a dodecahedron and an icosahedron, not necessarily to scale.
