MathDB

2

Part of 2015 South East Mathematical Olympiad

Problems(2)

Incenter diamter

Source: China Southeast Math Olympiad 2015 Day 1 P2

6/6/2017
Let II be the incenter of △ABC\triangle ABC with AB>ACAB>AC. Let Γ\Gamma be the circle with diameter AIAI. The circumcircle of △ABC\triangle ABC intersects Γ\Gamma at points A,DA,D, with point DD lying on \overarc{AC} (not containing BB). Let the line passing through AA and parallel to BCBC intersect Γ\Gamma at points A,EA,E. If DIDI is the angle bisector of ∠CDE\angle CDE, and ∠ABC=33∘\angle ABC = 33^{\circ}, find the value of ∠BAC\angle BAC.
geometryincenter
2015 China South East MO Grade 11 P2

Source: 2015 China South East MO Grade 11 P2

1/16/2018
Given a sequence {an}n∈Z+\{ a_n\}_{n\in \mathbb{Z}^+} defined by a1=1a_1=1 and a2k=a2k−1+ak,a2k+1=a2ka_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k} for all positive integer kk. Prove that, for any positive integer nn, a2n>2n24a_{2^n}>2^{\frac{n^2}{4}}.
inequalitiesalgebra