MathDB

2

Part of 2021 New Zealand MO

Problems(2)

bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2

Source:

9/20/2021
Let ABCDABCD be a trapezium such that AB∥CDAB\parallel CD. Let EE be the intersection of diagonals ACAC and BDBD. Suppose that AB=BEAB = BE and AC=DEAC = DE. Prove that the internal angle bisector of ∠BAC\angle BAC is perpendicular to ADAD.
geometryangle bisectortrapezoidperpendicular
x^2 +8/xy + y^2 >= 8 New Zealand MO 2021 Round 2 p2 NZMO

Source:

10/6/2021
Prove that x2+8xy+y2≥8x^2 +\frac{8}{xy}+ y^2 \ge 8 for all positive real numbers xx and yy.
algebrainequalities