MathDB

2

Part of 1988 Iran MO (2nd round)

Problems(2)

I, J and O are collinear - [Iran Second Round 1988]

Source:

12/7/2010
In a cyclic quadrilateral ABCDABCD, let I,JI,J be the midpoints of diagonals AC,BDAC, BD respectively and let OO be the center of the circle inscribed in ABCD.ABCD. Prove that I,JI, J and OO are collinear.
geometryconicscyclic quadrilateralgeometry proposed
Inequality on altitudes - [Iran Second Round 1988]

Source:

12/7/2010
In tetrahedron ABCDABCD let ha,hb,hch_a, h_b, h_c and hdh_d be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that 1ha<1hb+1hc+1hd.\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.
inequalitiesgeometry3D geometrytetrahedronPythagorean Theoremgeometry proposed