MathDB

8

Part of 2021 Tuymaada Olympiad

Problems(2)

Neat Geometric Inequality

Source: Tuymaada 2021 Senior P7

7/28/2021
An acute triangle ABCABC is given, AC≠BCAC \not= BC. The altitudes drawn from AA and BB meet at HH and intersect the external bisector of the angle CC at YY and XX respectively. The external bisector of the angle AHBAHB meets the segments AXAX and BYBY at PP and QQ respectively. If PX=QYPX = QY, prove that AP+BQ≥2CHAP + BQ \ge 2CH.
geometric inequalityinequalitiesgeometry
Sequence of trinomials?

Source: Tuymaada 2021 Senior P8

7/28/2021
In a sequence PnP_n of quadratic trinomials each trinomial, starting with the third, is the sum of the two preceding trinomials. The first two trinomials do not have common roots. Is it possible that PnP_n has an integral root for each nn?
polynomialquadraticsalgebrarational functions