2

Part of 2006 Turkey MO (2nd round)

Problems(2)

2006 students and 14 teachers - find maximum of t

Source: Turkey National Olympiad 2006 - D1 - P2

2006

14

teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least

t.

Find the maximum possible value of

t.

ratiograph theorycombinatorics unsolvedcombinatorics

turkey nmo 2006 q5

Source:

ABC

be a triangle. Its incircle touches the sides

CB, AC, AB

respectively at

N_{A},N_{B},N_{C}

. The orthic triangle of

ABC

H_{A}H_{B}H_{C}

H_{A}, H_{B}, H_{C}

are respectively on

BC, AC, AB

. The incenter of

AH_{C}H_{B}

I_{A}

I_{B}

I_{C}

were defined similarly. Prove that the hexagon

I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}

has all sides equal.

geometryincentercircumcirclegeometric transformationhomothetytrigonometrygeometry unsolved