MathDB

3

Part of 1997 Estonia National Olympiad

Problems(4)

computational in a circle

Source: 1997 Estonia National Olympiad Final Round grade 9

3/11/2020
The points A,B,MA, B, M and NN are on a circle with center OO such that the radii OAOA and OBOB are perpendicular to each other, and MNMN is parallel to ABAB and intersects the radius OAOA at PP. Find the radius of the circle if ∣MP∣=12|MP|= 12 and ∣PN∣=214|P N| = 2 \sqrt{14}
geometryradiuscircle
each diagonal of a convex pentagon is parallel to one of its sides, ratio

Source: 1997 Estonia National Olympiad Final Round grade 11 p3

3/11/2020
Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.
diagonalsparallelpentagongeometryratio
tan A : tan B : tan C =1:2:3 => AC/ AB= ?

Source: 1997 Estonia National Olympiad Final Round grade 10

3/11/2020
In triangle ABC, consider the sizes tan⁡∠A,tan⁡∠B\tan \angle A, \tan \angle B, and tan⁡∠C\tan \angle C into another such as the numbers 1,21, 2 and 33. Find the ratio of the sidelenghts ACAC and ABAB of the triangle.
ratiotrigonometrygeometry
ratio of volumes of 2 tetrahedra inscribed and circumscribed in sphere

Source: 1997 Estonia National Olympiad Final Round grade 12 p3

3/11/2020
A sphere is inscribed in a regular tetrahedron. Another regular tetrahedron is inscribed in the sphere. Find the ratio of the volumes of these two tetrahedra.
3D geometrytetrahedronVolumegeometrysphereinscribedcircumscribed