MathDB

3

Part of 2021 Dutch IMO TST

Problems(2)

reflection of K wrt OH lies on BC, orthocenter, 2 circumcenters

Source: 2021 Dutch IMO TST 1.3

12/29/2021
Let ABCABC be an acute-angled and non-isosceles triangle with orthocenter HH. Let OO be the center of the circumscribed circle of triangle ABCABC and let KK be center of the circumscribed circle of triangle AHOAHO. Prove that the reflection of KK wrt OHOH lies on BCBC.
geometryorthocentercollinearcircumcircle
n | 4a^2 + 9b^2 -1

Source: 2021 Dutch IMO TST 2.3

12/28/2021
Prove that for every positive integer nn there are positive integers aa and bb exist with n∣4a2+9b2−1n | 4a^2 + 9b^2 -1.
number theorydividesdivisible