MathDB

6

Part of 2013 Hanoi Open Mathematics Competitions

Problems(2)

scent of a classic area inequality (2013 HOMC Junior Q6)

Source:

7/18/2019
Let ABCABC be a triangle with area 11 (cm2^2). Points D,ED,E and FF lie on the sides AB,BCAB, BC and CA, respectively. Prove that min{min\{area of △ADF,\vartriangle ADF, area of △BED,\vartriangle BED, area of △CEF}≤14\vartriangle CEF\} \le \frac14 (cm2^2).
geometrygeometric inequalityareasarea of a triangle
C_n^{2013} > C_a^{2013}, combinations (HOMC 2013 Q6)

Source:

7/30/2019
Let be given a∈{0,1,2,3,...,100}.a\in\{0,1,2, 3,..., 100\}. Find all n∈{1,2,3,...,2013}n \in\{1,2, 3,..., 2013\} such that Cn2013>Ca2013C_n^{2013} > C_a^{2013} , where Ckm=m!k!(m−k)!C_k^m=\frac{m!}{k!(m -k)!}.
Combinationsinequalities