MathDB

3

Part of 2012 CentroAmerican

Problems(2)

Sum of 1/a+b = 1

Source: Centroamerican 2012, Problem 3

6/19/2012
Let a,b,ca,b,c be real numbers that satisfy 1a+b+1b+c+1a+c=1\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1 and ab+bc+ac>0ab+bc+ac >0.
Show that a+b+c−abcab+bc+ac≥4 a+b+c - \frac{abc}{ab+bc+ac} \ge 4
inequalitiesinequalities unsolved
Parallel to angle bisector

Source: Centroamerican 2012, Problem 6

6/20/2012
Let ABCABC be a triangle with AB<BCAB < BC, and let EE and FF be points in ACAC and ABAB such that BF=BC=CEBF = BC = CE, both on the same halfplane as AA with respect to BCBC.
Let GG be the intersection of BEBE and CFCF. Let HH be a point in the parallel through GG to ACAC such that HG=AFHG = AF (with HH and CC in opposite halfplanes with respect to BGBG). Show that ∠EHG=∠BAC2\angle EHG = \frac{\angle BAC}{2}.
geometryincenterparallelogramgeometry unsolved