MathDB

5

Part of 1998 IMO Shortlist

Problems(2)

IMO ShortList 1998, geometry problem 5

Source: IMO ShortList 1998, geometry problem 5

10/14/2004
Let ABCABC be a triangle, HH its orthocenter, OO its circumcenter, and RR its circumradius. Let DD be the reflection of the point AA across the line BCBC, let EE be the reflection of the point BB across the line CACA, and let FF be the reflection of the point CC across the line ABAB. Prove that the points DD, EE and FF are collinear if and only if OH=2ROH=2R.
geometrycircumcirclereflectionhomothetyparallelogramIMO Shortlistimo shortlist 1998
IMO ShortList 1998, number theory problem 5

Source: IMO ShortList 1998, number theory problem 5

10/22/2004
Determine all positive integers nn for which there exists an integer mm such that 2nāˆ’1{2^{n}-1} is a divisor of m2+9{m^{2}+9}.
modular arithmeticnumber theoryDivisibilityIMO Shortlist