MathDB

2

Part of 2010 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2010 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
In convex quadrilateral ABCDABCD, diagonals ACAC and BDBD intersect at point OO at angle 90∘90^{\circ}. Let KK, LL, MM and NN be orthogonal projections of point OO to sides ABAB, BCBC, CDCD and DADA of quadrilateral ABCDABCD. Prove that KLMNKLMN is cyclic
geometryCyclicquadrilateral
Regional Olympiad - FBH 2010 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
It is given acute triangle ABCABC with orthocenter at point HH. Prove that AH⋅ha+BH⋅hb+CH⋅hc=a2+b2+c22AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2} where aa, bb and cc are sides of a triangle, and hah_a, hbh_b and hch_c altitudes of ABCABC
geometryidentity
Regional Olympiad - FBH 2010 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2010

9/27/2018
Angle bisector from vertex AA of acute triangle ABCABC intersects side BCBC in point DD, and circumcircle of ABCABC in point EE (different from AA). Let FF and GG be foots of perpendiculars from point DD to sides ABAB and ACAC. Prove that area of quadrilateral AEFGAEFG is equal to the area of triangle ABCABC
geometryangle bisectorareacircumcircle