MathDB

6

Part of 2022 Novosibirsk Oral Olympiad in Geometry

Problems(3)

angle bisector wanted in 120-40-20 (2022 Novosibirsk Oral Geo Oly 7.6)

Source:

9/2/2023
A triangle ABCABC is given in which ∠BAC=40o\angle BAC = 40^o. and ∠ABC=20o\angle ABC = 20^o. Find the length of the angle bisector drawn from the vertex CC, if it is known that the sides ABAB and BCBC differ by 44 centimeters.
geometryangle bisector
cut isosceles rigth triangle in 9 triangles (2022 Novosibirsk Oral Geo Oly 8.6)

Source:

9/2/2023
Anton has an isosceles right triangle, which he wants to cut into 99 triangular parts in the way shown in the picture. What is the largest number of the resulting 99 parts that can be equilateral triangles?
A more formal description of partitioning. Let triangle ABCABC be given. We choose two points on its sides so that they go in the order AC1C2BA1A2CB1B2AC_1C_2BA_1A_2CB_1B_2, and no two coincide. In addition, the segments C1A2C_1A_2, A1B2A_1B_2 and B1C2B_1C_2 must intersect at one point. Then the partition is given by segments C1A2C_1A_2, A1B2A_1B_2, B1C2B_1C_2, A1C2A_1C_2, B1A2B_1A_2 and C1B2C_1B_2. https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png
geometrycombinatorial geometryright triangleisoscelespartitions
equilaterals AX/XB =BY/YC = CZ/ZA = 2/1 (2022 Novosibirsk Oral Geo Oly 9.6)

Source:

9/3/2023
Triangle ABCABC is given. On its sides ABAB, BCBC and CACA, respectively, points X,Y,ZX, Y, Z are chosen so that AX:XB=BY:YC=CZ:ZA=2:1.AX : XB =BY : YC = CZ : ZA = 2:1. It turned out that the triangle XYZXYZ is equilateral. Prove that the original triangle ABCABC is also equilateral.
geometryEquilateralequilaterals