MathDB

6

Part of 2007 Sharygin Geometry Olympiad

Problems(4)

number of symmetry axes of a checked polygon and polyhedron

Source: 2007 Sharygin Geometry Olympiad Correspondence Round P6

4/25/2019
a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?
geometrysymmetryaxespolygonpolyhedron
analogous triangles but not congruent

Source: Sharygin Final 2007 8.6

4/30/2019
Two non-congruent triangles are called analogous if they can be denoted as ABCABC and A′B′C′A'B'C' such that AB=A′B′,AC=A′C′AB = A'B', AC = A'C' and ∠B=∠B′\angle B = \angle B' . Do there exist three mutually analogous triangles?
geometrycongruent trianglesequal angles
dissecting a cube (2n+1)^2 into 1x1x1 and 2x2x1, min no of small cubes

Source: Sharygin Final 2007 9.6

4/30/2019
A cube with edge length 2n+12n+ 1 is dissected into small cubes of size 1×1×11\times 1\times 1 and bars of size 2×2×12\times 2\times 1. Find the least possible number of cubes in such a dissection.
geometrycombinatorial geometry
6 circles, other concentric other tangent, 8 intersections lie in 2 circles

Source: Sharygin 2007 Final 10.6

4/30/2019
Given are two concentric circles Ω\Omega and ω\omega. Each of the circles b1b_1 and b2b_2 is externally tangent to ω\omega and internally tangent to Ω\Omega, and ω\omega each of the circles c1c_1 and c2c_2 is internally tangent to both Ω\Omega and ω\omega. Mark each point where one of the circles b1,b2b_1, b_2 intersects one of the circles c1c_1 and c2c_2. Prove that there exist two circles distinct from b1,b2,c1,c2b_1, b_2, c_1, c_2 which contain all 88 marked points. (Some of these new circles may appear to be lines.)
circlesgeometrycombinatorial geometry