MathDB

1

Part of 2005 IMAR Test

Problems(2)

abc \geq 1 then \sum \frac 1{1+b+c} \leq 1

Source: GMB-IMAR 2005, Juniors, Problem 1

10/10/2005
Let a,b,ca,b,c be positive real numbers such that abc≥1abc\geq 1. Prove that 11+b+c+11+c+a+11+a+b≤1. \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5.
AMCUSAMOInequalitythree variable inequality
Some incircles...

Source: GMB-IMAR 2005, Seniors, Problem 1

10/9/2005
The incircle of triangle ABCABC touches the sides BC,CA,ABBC,CA,AB at the points D,E,FD,E,F, respectively. Let KK be a point on the side BCBC and let MM be the point on the line segment AKAK such that AM=AE=AFAM=AE=AF. Denote by L,NL,N the incenters of triangles ABK,ACKABK,ACK, respectively. Prove that KK is the foot of the altitude from AA if and only if DLMNDLMN is a square. [hide="Remark"]This problem is slightly connected to [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=344774#p344774]GMB-IMAR 2005, Juniors, Problem 2 Bogdan Enescu
geometryincenterrectanglecircumcircleperpendicular bisectorgeometry proposed