MathDB

2

Part of 2016 Latvia National Olympiad

Problems(4)

2016 Latvia National Olympiad 3rd Round Grade9Problem2

Source:

7/22/2016
Triangle ABCABC has median AFAF, and DD is the midpoint of the median. Line CDCD intersects ABAB in EE. Prove that BD=BFBD = BF implies AE=DEAE = DE!
geometry
2016 Latvia National Olympiad 3rd Round Grade11Problem2

Source:

7/22/2016
An acute triangle ABCABC (AB>ACAB > AC) has circumcenter OO, but DD is the midpoint of BCBC. Circle with diameter ADAD intersects sides ABAB and ACAC in EE and FF respectively. On segment EFEF pick a point MM so that DM∥AODM \parallel AO. Prove that triangles ABDABD and FDMFDM are similar.
geometrycircumcircle
2016 Latvia National Olympiad 3rd Round Grade10Problem2

Source:

7/22/2016
The bisectors of the angles ∢CAB\sphericalangle CAB and ∢BCA\sphericalangle BCA intersect the circumcircle of ABCABC in PP and QQ respectively. These bisectors intersect each other in point II. Prove that PQ⊥BIPQ \perp BI.
geometrycircumcircle
2016 Latvia National Olympiad 3rd Round Grade12Problem2

Source:

7/22/2016
Triangle ABCABC has incircle ω\omega and incenter II. On its sides ABAB and BCBC we pick points PP and QQ respectively, so that PI=QIPI = QI and PB>QBPB > QB. Line segment QIQI intersects ω\omega in TT. Draw a tangent line to ω\omega passing through TT; it intersects the sides ABAB and BCBC in UU and VV respectively. Prove that PU=UV+VQPU = UV + VQ!
geometryincenter