MathDB

3

Part of 2016 Danube Mathematical Olympiad

Problems(2)

Smallest Value Of An Angle

Source: Mathematical Danube Competition 2016, Juniors P3

4/21/2022
Let ABCABC be a triangle with AB<AC,AB < AC, II its incenter, and MM the midpoint of the side BCBC. If IA=IM,IA=IM, determine the smallest possible value of the angle AIMAIM.
geometryromania
Problem 3, Inequality with Integral

Source:

10/29/2017
3. Let n > 1 be an integer and a1,a2,...,ana_1, a_2, . . . , a_n be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that ∑ak1+(a0+a1+...+ak−1)2≥c\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c, where a0=0a_0 = 0. b) Show that ’the best’ value of c is at least π4\frac{\pi}{4}.
inequalitiescalculusintegration