MathDB

g8

Part of Indonesia MO Shortlist - geometry

Problems(4)

only 1 triangle with sides consecutive naturals and <A=2<B

Source: Indonesia INAMO Shortlist 2008 G8

8/25/2021
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
geometryconsecutiveequal anglesIntegers
AD/AP_1+ BE/BP_2 + CF/CP_3 >=6 , angle bisectors

Source: Indonesia INAMO Shortlist 2009 G8 https://artofproblemsolving.com/community/c1101409_

12/10/2021
Suppose the points D,E,FD, E, F lie on sides BC,CA,ABBC, CA, AB, respectively, so that AD,BE,CFAD, BE, CF are angle bisectors. Define P1P_1, P2P_2, P3P_3 respectively as the intersection point of ADAD with EFEF, BEBE with DFDF, CFCF with DEDE respectively. Prove that ADAP1+BEBP2+CFCP3≥6\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6
geometrygeometric inequalityAngle Bisectorsangle bisector
altitudes of ABC if angle bisectors of DEF, converse of incenter property

Source: Indonesia INAMO Shortlist 2010 G8

8/27/2021
Given an acute triangle ABCABC and points DD, EE, FF on sides BCBC, CACA and ABAB, respectively. If the lines DADA, EBEB and FCFC are the angle bisectors of triangle DEFDEF, prove that the three lines are the altitudes of triangle ABCABC.
geometryangle bisectoralttiudesincenter
concyclic wanted , 2 diameters, 3 circles related

Source: Indonesia INAMO Shortlist 2017 G8 https://artofproblemsolving.com/community/c1101409_indonesia_shortlist__geometry

11/15/2021
Given a circle centered at point OO, with ABAB as the diameter. Point CC lies on the extension of line ABAB so that BB lies between AA and CC, and the line through CC intersects the circle at points DD and EE (where DD lies between CC and EE). OFOF is the diameter of the circumcircle of triangle OBDOBD, and the extension of the line CFCF intersects the circumcircle of triangle OBDOBD at point GG. Prove that the points O,A,E,GO, A, E, G lie on a circle.
Concyclicgeometry