MathDB

2

Part of 2015 China Second Round Olympiad

Problems(2)

Element belonging to at least n/k subsets

Source: China Second Round 2015 (A) Q2

5/5/2016
Let S={A1,A2,…,An}S=\{A_1,A_2,\ldots ,A_n\}, where A1,A2,…,AnA_1,A_2,\ldots ,A_n are nn pairwise distinct finite sets (n≥2)(n\ge 2), such that for any Ai,Aj∈SA_i,A_j\in S, Ai∪Aj∈SA_i\cup A_j\in S. If k=min⁡1≤i≤n∣Ai∣≥2k= \min_{1\le i\le n}|A_i|\ge 2, prove that there exist x∈⋃i=1nAix\in \bigcup_{i=1}^n A_i, such that xx is in at least nk\frac{n}{k} of the sets A1,A2,…,AnA_1,A_2,\ldots ,A_n (Here ∣X∣|X| denotes the number of elements in finite set XX).
combinatoricsset theory
Prove CD^2=BDxCE

Source: China Second Round 2015 (B) Q2

5/5/2016
In isoceles △ABC\triangle ABC, AB=ACAB=AC, II is its incenter, DD is a point inside △ABC\triangle ABC such that I,B,C,DI,B,C,D are concyclic. The line through CC parallel to BDBD meets ADAD at EE. Prove that CD2=BD⋅CECD^2=BD\cdot CE.
geometryincenter