MathDB

2

Part of 2023 New Zealand MO

Problems(2)

|DAQ|^2 = |PAQ| x |BCD| , ABCD #

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p2

9/2/2023
Let ABCDABCD be a parallelogram, and let PP be a point on the side ABAB. Let the line through PP parallel to BCBC intersect the diagonal ACAC at point QQ. Prove that ∣DAQ∣2=∣PAQ∣×∣BCD∣,|DAQ|^2 = |PAQ| \times |BCD| , where ∣XYZ∣|XY Z| denotes the area of triangle XYZXY Z.
parallelogramareasarea of a trianglegeometry
at least one of a,b,c >17/10 if a,b,c>0 with a+b+c = abc

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p2

9/2/2023
Let a,ba, b and cc be positive real numbers such that a+b+c=abca+b+c = abc. Prove that at least one of a,ba, b or cc is greater than 1710\frac{17}{10} .
algebrainequalities