MathDB

7

Part of 2020 Novosibirsk Oral Olympiad in Geometry

Problems(3)

min n, n equal triangles in convex n-gon (2020 Novosibirsk Oral Geo Oly 7.7)

Source:

4/27/2021
The segments connecting the interior point of a convex non-sided nn-gon with its vertices divide the nn-gon into nn congruent triangles. For what is the smallest nn that is possible?
geometrycombinatorial geometrycongruent triangles
bisector wanted, 30^o, 150^o, AB=BD (2020 Novosibirsk Oral Geo Oly 8.7)

Source:

5/20/2021
You are given a quadrilateral ABCDABCD. It is known that ∠BAC=30o\angle BAC = 30^o, ∠D=150o\angle D = 150^o and, in addition, AB=BDAB = BD. Prove that ACAC is the bisector of angle CC.
geometryangle bisectorangles
AD=AB+CD, tangent circle to 3 sides, cyclic (2020 Novosibirsk Oral Geo Oly 9.7)

Source:

5/24/2021
The quadrilateral ABCDABCD is known to be inscribed in a circle, and that there is a circle with center on side ADAD tangent to the other three sides. Prove that AD=AB+CDAD = AB + CD.
geometryCyclicTangents