MathDB

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Part of 2015 India Regional MathematicaI Olympiad

Problems(5)

angle chasing in RMO, cyclic ABCD, 2 circumcircles, incenter, right wanted

Source: CRMO 2015 region 1 p1

9/30/2018
In a cyclic quadrilateral ABCDABCD, let the diagonals ACAC and BDBD intersect at XX. Let the circumcircles of triangles AXDAXD and BXCBXC intersect again at YY . If XX is the incentre of triangle ABYABY , show that ∠CAD=90o\angle CAD = 90^o.
geometrycircumcircleincenterright anglecyclic quadrilateral
Question 1

Source:

12/6/2015
Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.
geometrygeometric transformationreflectionangle bisector
RMO Delhi Q.1

Source:

12/31/2015
2 circles Γ and Σ, with centers O and P, respectively, are such that P lies on Γ. Let A be a point on Σ, and let M be the midpoint of AP. Let B be another point on Σ, such that AB||OM. Then prove that the midpoint of AB lies on Γ.
RMO 2015 Karnataka geometry, circumcenter lies on an angle bisector

Source: CRMO 2015 region 4 (Karnataka) p1

9/30/2018
Let ABCABC be a triangle. Let B′B' denote the reflection of bb in the internal angle bisector ll of ∠A\angle A.Show that the circumcentre of the triangle CB′ICB'I lies on the line ll where II is the incentre of ABCABC.
geometryCircumcenterincenterreflection
Condition on sides of a quadrilateral and some diagonals

Source: RMO (Mumbai Region) 2015 P1

12/6/2015
Let ABCDABCD be a convex quadrilateral with AB=aAB=a, BC=bBC=b, CD=cCD=c and DA=dDA=d. Suppose a2+b2+c2+d2=ab+bc+cd+da,a^2+b^2+c^2+d^2=ab+bc+cd+da, and the area of ABCDABCD is 6060 sq. units. If the length of one of the diagonals is 3030 units, determine the length of the other diagonal.
inequalitiesinequalities proposedrearrangement inequalityAM-GMgeometrygeometry proposed