MathDB

4

Part of 1989 Greece National Olympiad

Problems(3)

2|3-2x|-|x-2|=x 1989 Greece MO Grade X p4

Source:

9/7/2024
Solve 2∣3−2x∣−∣x−2∣=x2|3-2x|-|x-2|=x , x∈Rx\in\mathbb{R} .
algebra
h^2 <= ab for tangential trapezoid

Source: 1989 Greece MO Grade XI p4

9/7/2024
A trapezoid with bases a,ba,b and altitude hh is circumscribed around a circl.. Prove that h2≤abh^2\le ab.
geometrytrapezoidtangentialgeometric inequalityGeometric Inequalities
(x^ky)^2=e if x^{n}=e,y^2=e,yxy=x^{-1}

Source: 1989 Greece MO Grade XII p4

9/7/2024
In a group GG, we have two elements x,yx,y such that xn=e,y2=e,yxy=x−1x^{n}=e,y^2=e,yxy=x^{-1}, n≥1n\ge 1. Prove that for any k∈Nk\in\mathbb{N} holds (xky)2=e(x^ky)^2=e.
Note : e=group's identity .
group theoryalgebrasuperior algebra