MathDB

2

Part of 2002 Hungary-Israel Binational

Problems(2)

p(M ) = \frac{MA′ . MB′ . MC ′ }{MA. MB.MC}

Source: 13-th Hungary-Israel Binational Mathematical Competition 2002

4/7/2007
Let A′,B′,C′A', B' , C' be the projections of a point MM inside a triangle ABCABC onto the sides BC,CA,ABBC, CA, AB, respectively. Define p(M)=MA′⋅MB′⋅MC′MA⋅MB⋅MCp(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC} . Find the position of point MM that maximizes p(M)p(M ).
trigonometrygeometryincenterinequalitiesgeometry proposed
Find the angles of a triangle

Source: 13-th Hungary-Israel Binational Mathematical Competition 2002

4/7/2007
Points A1,B1,C1A_{1}, B_{1}, C_{1} are given inside an equilateral triangle ABCABC such that B1AB^=A1BA^=150,C1BC^=B1CB^=200,A1CA^=C1AC^=250\widehat{B_{1}AB}= \widehat{A1BA}= 15^{0}, \widehat{C_{1}BC}= \widehat{B_{1}CB}= 20^{0}, \widehat{A_{1}CA}= \widehat{C_{1}AC}= 25^{0}. Find the angles of triangle A1B1C1A_{1}B_{1}C_{1}.
geometry proposedgeometry