MathDB

11.5

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

parallelogram by 4 similar isosceles triangles

Source: III Soros Olympiad 1996-97 R1 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
The area of a convex quadrilateral is SS, and the angle between the diagonals is aa. On the sides of this quadrilateral, as on the bases, isosceles triangles with vertex angle equal to ϕ\phi, wherein two opposite triangles are located on the other side of the corresponding side of the quadrilateral than the quadrilateral itself, and the other two are located on the other side. Prove that the vertices of the constructed triangles, different from the vertices of the quadrilateral, serve as the vertices of a parallelogram. Find the area of this parallelogram.
geometryparallelogram
folding a paper triangle to get surface of a regular unit tetradragon

Source: III Soros Olympiad 1996-97 R2 11. 5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
Prove that this triangle cut out of paper can be folded so that the surface of a regular unit tetradragon (i.e., a triangular pyramid, all edges of which are equal to 11) is obtained if: a) this triangle is isosceles, the lateral sides are equal to 22 , the angle between them is 120o120^o, b) two sides of this triangle are equal to 22 and 232\sqrt3, the angle between them is 150o150^o.
geometrycombinatorial geometry
plane tangent to insphere of parallelepiped

Source: III Soros Olympiad 1996-97 R3 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/31/2024
All faces of the parallelepiped ABCDA1B1C1D1ABCDA_1B_1C_1D_1 are equal rhombuses. Plane angles at vertex AA are equal. Points KK and MM are taken on the edges A1B1A_1B_1 and A1D1A_1D_1. It is known that A1K=aA_1K = a, A1M=bA_1M = b, anda+b a + b is an edge of the parallelepiped. Prove that the plane AKMAKM touches the sphere inscribed in the parallelepiped. Let us denote by QQ the touchpoint of this sphere with the plane AKMAKM . In what ratio does the straight line AQAQ divide the segment KMKM?
geometryparallelepiped3D geometry