MathDB

2

Part of 2007 Serbia National Math Olympiad

Problems(2)

a triangle is dissected into $25$ small triangle...

Source: Serbian Mathematical Olympiad 2007

6/10/2007
Triangle ΔGRB\Delta GRB is dissected into 2525 small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex GG is painted in green, vertex RR in red, and BB in blue; Each vertex on side GRGR is either green or red, each vertex on RBRB is either red or blue, and each vertex on GBGB is either green or blue. The vertices inside the big triangle are arbitrarily colored. Prove that, regardless of the way of coloring, at least one of the 2525 small triangles has vertices of three different colors.
combinatorics proposedcombinatorics
intersect on the incircle

Source: Serbian Mathematical Olympiad 2007

6/9/2007
In a scalene triangle ABC,AD,BE,CFABC , AD, BE , CF are the angle bisectors (D∈BC,E∈AC,F∈AB)(D \in BC , E \in AC , F \in AB). Points Ka,Kb,KcK_{a}, K_{b}, K_{c} on the incircle of triangle ABCABC are such that DKa,EKb,FKcDK_{a}, EK_{b}, FK_{c} are tangent to the incircle and Ka∉BC,Kb∉AC,Kc∉ABK_{a}\not\in BC , K_{b}\not\in AC , K_{c}\not\in AB. Let A1,B1,C1A_{1}, B_{1}, C_{1} be the midpoints of sides BC,CA,ABBC , CA, AB , respectively. Prove that the lines A1Ka,B1Kb,C1KcA_{1}K_{a}, B_{1}K_{b}, C_{1}K_{c} intersect on the incircle of triangle ABCABC.
geometryprojective geometrycomplex numberscyclic quadrilateralgeometry proposed