MathDB

1

Part of 2018 India Regional Mathematical Olympiad

Problems(2)

GCD of sides

Source: RMO 2018 P1

10/7/2018
Let ABCABC be a triangle with integer sides in which AB<ACAB<AC. Let the tangent to the circumcircle of triangle ABCABC at AA intersect the line BCBC at DD. Suppose ADAD is also an integer. Prove that gcd⁡(AB,AC)>1\gcd(AB,AC)>1.
greatest common divisorgeometrycircumcircle
Again a geometry in RMO

Source: RMO 2018 P1

10/28/2018
Let ABCABC be an acute angled triangle and let DD be an interior point of the segment BCBC. Let the circumcircle of ACDACD intersect ABAB at EE (EE between AA and BB) and let circumcircle of ABDABD intersect ACAC at FF (FF between AA and CC). Let OO be the circumcenter of AEFAEF. Prove that ODOD bisects ∠EDF\angle EDF.
geometrycircumcircleangle bisector