MathDB

2

Part of 2016 Bulgaria JBMO TST

Problems(2)

Inequality

Source: Bulgaria JTST 2016 P2 day 1

8/27/2019
a, b, c are positive real numbers and a+b+c=k. Find the minimum value of b2/(ka+bc)1/2+c2/(kb+ac)1/2+a2/(kc+ab)1/2 b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2
Inequality proposedinequalities
quadrilateral of 4 orthocenters in a cyclic pentagon is square iff criterion

Source: 2016 Bulgaria JBMO TST 2.2

8/11/2020
The vertices of the pentagon ABCDEABCDE are on a circle, and the points H1,H2,H3,H4H_1, H_2, H_3,H_4 are the orthocenters of the triangles ABC,ABE,ACD,ADEABC, ABE, ACD, ADE respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if BE∥CDBE \parallel CD and the distance between them is BE+CD2\frac{BE + CD}{2}.
geometrysquareorthocenterCyclicpentagon