MathDB

1

Part of 1990 Greece National Olympiad

Problems(3)

a=b if |a-b|<x for any x>0 1990 Greece MO Grade X p1

Source:

9/6/2024
Let a,ba,b, be two real numbers. If for any x>0x>0 holds that ∣a−b∣<x|a-b|<x, then prove that a=ba=b.
algebrainequalities
BD/CD&lt;HD/MD in right triangle, altitude, bisector, median from A

Source: 1990 Greece MO Grade XI p1

9/6/2024
Let ABCABC be a right triangle with ∠A=90o\angle A=90^o and AB<ACAB<AC. Let AH,AD,AMAH,AD,AM be altitude, angle bisector and median respectively. Prove that BDCD<HDMD.\frac{BD}{CD}<\frac{HD}{MD}.
geometryangle bisectorgeometric inequalityright triangle
A^2=0 if A^3=0 for 2x2 real matric A

Source: 1990 Greece MO Grade XII p1

9/6/2024
Let AA be a 2 x 22\,x\,2 matrix with real numbers. Prove that if A3=OA^3=\mathbb{O} then A2=OA^2=\mathbb{O}.
linear algebramatrix