MathDB

3

Part of 1999 Bundeswettbewerb Mathematik

Problems(2)

3 isosceles given on a line with an angle 120, equilateral wanted

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

1/27/2020
In the plane are given a segment ACAC and a point BB on the segment. Let us draw the positively oriented isosceles triangles ABS1,BCS2ABS_1, BCS_2, and CAS3CAS_3 with the angles at S1,S2,S3S_1,S_2,S_3 equal to 120o120^o. Prove that the triangle S1S2S3S_1S_2S_3 is equilateral.
geometryisoscelesEquilateral
\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}, parallelograms inscribed a convex ABCD

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

1/27/2020
Let PP be a point inside a convex quadrilateral ABCDABCD. Points K,L,M,NK,L,M,N are given on the sides AB,BC,CD,DAAB,BC,CD,DA respectively such that PKBLPKBL and PMDNPMDN are parallelograms. Let S,S1S,S_1, and S2S_2 be the areas of ABCD,PNAKABCD, PNAK, and PLCMPLCM. Prove that S≥S1+S2\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}.
geometryparallelogramconvex quadrilateralarea of a trianglegeometric inequality