MathDB

10.6

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

angle chasing, angle bisectors (V Soros Olympiad 1998-99 Round 1 10.6)

Source:

5/25/2024
In triangle ABCABC, the bisectors of the internal angles AA1AA_1 , BB1BB_1 and CC1CC_1 are drawn (A1,B1A_1, B_1, C1C_1 - on the sides of the triangle). It is known that ∠AA1C=∠AC1B1\angle AA_1C = \angle AC_1B_1. Find ∠BCA\angle BCA.
geometryangle bisectorangles
angle chasing, OI line (V Soros Olympiad 1998-99 Round 2 10.6)

Source:

5/25/2024
The straight line containing the centers of the circumscribed and inscribed circles of triangle ABCABC intersects rays BABA and BCBC and forms an angle with the altitude to side BCBC equal to half the angle ∠BAC\angle BAC. What is angle ∠ABC\angle ABC?
geometryangles
a_{n+1} = 3a_n-2a_{n-1} (V Soros Olympiad 1998-99 Round 3 10.6)

Source:

5/26/2024
Find the formula for the general term of the sequence an, for which a1=1a_1 = 1, a2=3a_2 = 3, an+1=3an−2an−1a_{n+1} = 3a_n-2a_{n-1} (you need to express an in terms of nn).
algebraSequenceRecurrencerecurrence relations