MathDB

2

Part of 1995 Iran MO (2nd round)

Problems(2)

The incenter lies on circumcircle [Iran Second Round 95]

Source:

11/25/2010
Let ABCABC be an acute triangle and let ℓ\ell be a line in the plane of triangle ABC.ABC. We've drawn the reflection of the line ℓ\ell over the sides AB,BCAB, BC and ACAC and they intersect in the points A′,B′A', B' and C′.C'. Prove that the incenter of the triangle A′B′C′A'B'C' lies on the circumcircle of the triangle ABC.ABC.
geometryincentercircumcirclegeometric transformationreflectiongeometry proposed
Ceil function [Iran Second Round 1995]

Source:

11/25/2010
Let n≥0n \geq 0 be an integer. Prove that ⌈n+n+1+n+2⌉=⌈9n+8⌉ \lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil \sqrt{9n+8} \rceil Where ⌈x⌉\lceil x \rceil is the smallest integer which is greater or equal to x.x.
functionceiling functionquadraticsnumber theory proposednumber theory